Alternatively, you can choose a coordinate system where the particle is rotating, accelerating, or moving however you like. Your choice will be determined by the goals you have. (Without the laws of motion, there is no reason to prefer even intertial frames over nonintertial ones.)
Is the particle moving or is the coordinate system?
It seems, like you say, to be arbitrary. Without something else in the universe, the particle could have any rotation, velocity, inertia, etc... depending on which coordinate system you choose. The coordinate system seems to be layered on, from a universe where the idea of space and time were more useful.
No, according to the Copenhagen interpretation.
Note that in general relativity, which allows inertial and non-inertial reference frames as equally valid, there is some non-arbitrary motion of objects that is independent of the choice of frame. These are the so-called tidal gravitational effects that cannot be transformed away in any frame, and are easiest described as curvature of spacetime due to stress-energy.
A smaller point: "motion" and "tidal...effects" each implicitly chooses a splitting (in particular a 1+3 threading) of spacetime into space and time. That's fine; it's really hard for general relativity to confront observation without doing some splitting, and implicitly threading is standard behaviour (Synge does this in his textbook, for example, when discussing 3-velocity and spatial distance and offering up his world function biscalar; note that this is different from explicitly applying a 1+3 formalism).
Your central point is right though: coordinate transformations or arbitrary choices of splittings -- 1+3 threading or 3+1 slicings are examples -- cannot get rid of real curvature.
A test particle is one with negligible mass and extent, and thus does not itself perturb the curvature of the background vacuum spacetime.
If a real particle is electrically neutral, extremely low mass, and nearly pointlike, then in a vacuum spacetime it should behave very much like a test particle. Moreover, for microscopic real objects, which aren't truly pointlike and which may not be wholly chargeless, it is impractically hard to tell at a large enough spatial distance that it's not a test particle.
So a test particle or test field does not source the spacetime curvature -- it just rides along the background spacetime, which is arbitrarily curved by its nature.
We can extend the notion to a test field that is almost wholly massless, like a single-electron quantum field. The test field sits on the background, but doesn't influence the background's dynamics.
Real quantum fields, like test fields, permeate the whole of spacetime. However they tend to have abundant stress-energy in them, so source some of the curvature of the whole spacetime, and typically not uniformly. One can fix a gauge (like how one calibrates an air-pressure altimeter) that lets one talk about the inherent curvature of the background spacetime and the contribution to the total curvature of the lowest density of the quantum field. Typically the background is chosen to be some exact vacuum solution of the EFEs with corrections added via something like perturbation theory; flat spacetime, Schwarzschild and Robertson-Walker expanding spacetimes are popular backgrounds, depending on what one is studying.
Sometimes, though one wants to think about an electron rather than some characteristic set of numbers in some all-over-spacetime fields. In that case one (probably implicitly) fixes a gauge that lets one talk about the electron as an independent object that generates an electromagnetic field which, because that has some nonzero stress-energy (zero being set by our gauging choices and being found sufficiently far from our electron) it influences the background curvature of the spacetime (the background again being set by our gauging choices).
In either paradigm, one might want to think about, in reverse order, the gravitational back-reaction of the electron (and its EM field)'s gravitational footprint on itself, or the gravitational back-reaction of the overdensities in the all-over-spacetime (classical or quantum) fields on the regions of spacetime they are found in. "Gravity gravitates" is the problem: this arises from the EFEs' non-linearity. Fortunately, for a single electron these back-reactions will be very small, because electrons in either paradigm have very little stress-energy, so their contribution to any curvature will thus be so extremely tiny that it will be practically immeasurable, so there is basically no hope of measuring higher-order contributions (that is, the electron's gravitational influence's own influence on the curvature). Negiligibility saves a lot of headaches, because working with gravitational backreaction is hard.
But to add to the first paragraph of their answer: if you set the 'T' tensor to the zero tensor in https://en.wikipedia.org/wiki/Einstein_field_equations#Mathe..., then you are effectively studying the vacuum. The left hand side describes the curvature of spacetime. This is a set of 10 non-linear partial differential equations which permit many solutions.
You can then add a "single particle" back to these solutions with varying degrees of complexity as outlined by raatgift.
It seems that the single particle may not be alone in the universe and that there is a vacuum entity that the particle could be depending upon. I'm not sure what forces or other properties the vacuum imbues the particle. Traveling along a curved spacetime wouldn't imply a force on a neutral particle, would it? I'm curious what properties of the particle the vacuum would change over time. I may have to dig deeper into the math to find out. I don't want to burden anyone with my questions.
My lay observation was that many of these properties are built up from relations (functions) between two points. The equations for distance have two points, velocity depends on distance over time, inertia depends on velocity. It's difficult to imagine what the other point would be in a single particle universe. It's why I wonder if there would be inertia in a universe with one particle. These concepts seem to spring forth once you add another particle, which causes me to think there isn't inertia per se, but a relation we observe that we call inertia under the condition of having a point moving relative to another with a certain mass at a certain speed. Useful no doubt, except perhaps in single particle universes. Similarly, if we have a party balloon and we take away all the gas and balloons in the universe except for one molecule of nitrogen, do we still have pressure? My answer here is not really, not practically, because these forces spring forth from the context of having multiple molecules. It's a helpful concept for use in the case balloons, tires and pipes, but there doesn't appear to be such a thing as pressure that exists in the universe independent of the particles within it.
> Travelling along a curved spacetime wouldn't imply a force on a neutral particle, would it?
There are exact solutions of the Einstein Field Equations (although typically this is done with the linearized version of them, the LEFEs, which is in the "almost certainly a good physical approximation" bucket) in which a change in the background metric propagates through vacuum in some direction as a plane wave.
The background metric could in principle be, for example, flat spacetime, or Schwarzschild, or any other vacuum solution of the EFEs. When one chooses a splitting of spacetime into space and time, the wave propagates through space.
A small set of test particles reacts to the wave as it passes over them, changing their respective "radar distances" for example. This view lends itself well to perturbation theory, where the wavelike perturbation of the background metric can (if weak) be quantized; those quanta are gravitons in perturbatively quantized gravity. That's a real theory that exists and is good everywhere where there is no strong gravity. "Strong gravity" can even be defined in terms of perturbatively quantized gravity as at least one loop of gravitons on a Feynman diagram. You generally don't find that until very close to a gravitational singularity (i.e., perturbatively quantized gravity as an effective field theory works inside stellar BH horizons too, and works even better inside even more massive BH horizons because there you are further away from the strong gravity near the centre.
In the spacetime view, the particles are instead worldlines that are mostly in regions described exactly by the background metric, but a small section of the worldlines are found in a small region where the metric is different. One can consider the spacetime intervals of points on the various worldlines and look for some minimum which will ought to be found in the region with that different metric. Exactly oppositely from Euclidean geometry, curved lines are shorter than straight lines, so a(n inertial) worldline which is found only in flat spacetime regions will be longer than one which is (even briefly) also found in a region with the slightly different metric.
Are such worldines travelling along? Nothing really travels at all if it's a worldline in a "block universe" spacetime picture like in the paragraph immediately above. But if one foliates spacetime into spaces with a time coordinate, movement emerges. You can choose a foliation and set of coordinates in which the different metric's effects are shoved into time-like effects (if the background is flat spacetime, a particle exposed to the gravitational wave is younger than a particle that isn't; c.f. the twin paradox); or you can choose a foliation and set of coordinates in which the different metric's effects are shoved into space-like ones ("radar" showing that the particle exposed to the gravitational wave is dragged or bobbed by its passage).
With some of these equivalent and (mostly) convertible views of the same physical picture, it would be perfectly reasonable to say that a force has been exerted. After all, one is interested in what causes a small piece of a long stable worldline to look different. The force acting on that section of worldline (or on the particle during that brief period of time), however, depends on one's choices of foliation, coordinates, and a few other things. If you can make some set of such choices such that the force vanishes, we call it an inertial force, d'Alembert force, or frame-dependent/fictitious force. Real forces cannot be made to vanish by any choice of coordinates etc.; real forces' "fingerprints" must do more than change an object's position or orientation in spacetime.
The tendency of objects -- particles, worldines, extended (many-particle) objects, worldtubes, or the full fields in which these worldtubes are just clusters of correlated numers -- to look mostly the same is one way of thinking about inertia: you'd want to attribute any unusual change to some force acting at the point of the change.
If you remove everything but your test object, so that no other object or field can impart any non-d'Alembertian force on it, then a magic observer at an enormous distance watching its behaviour could say that any changes in radar distance or (if the test object is a clock) ticking rate is the result of coupling with curved spacetime. But a magic observer who can see the whole worldline of the object and knows the layout of spacetime curvature will note that it matches a longest-possible solution of the geodesic equations.
> I don't want to burden anyone with my questions.
It's not a burden!
> there doesn't appear to be such a thing as pressure that exists in the universe independent of the particles within it
This is the thing about fixing a gauge: you've decided here that the "standard" pressure is vacuum. That's perfectly fine. But it's also perfectly fine to set the 0 point higher, and talk about negative pressures.
We do that with temperature, for instance; we set the 0 point of our temperature gauges and talk about positive (or positive and negative for e.g. celsius and fahernheit) temperatures. We do that with calendars, for instance, talking about years before or after some 0 point. We do that unthinkingly when we talk about "a quarter past [some hour]" or "a quarter to [some hour]", for instance.
Gauging is done -- often implicitly -- on a huge range of physical quantities. We can even formalize the process into a gauge theory, wherein we encode such choices into a gauge field, which can have its own dynamics. (As an example, let's make a field throughout the Earth's atmosphere where every point has a value which would be reported by a calibrated barometric pressure sensor; we can choose one particular reading as a surface and call it "FL10" and fly airplanes in that surface/layer. Depending on latitude and local weather conditions, "FL10" will coincide or deviate from the 10000 feet above the WGS84 surface reported by GPS equipment at the same point. Further depending on surface features, a radar altimeter may report anything from ~ 10000 feet to "oops you've just crashed into a mountain". Moreover a pressure altimeter can slide "QNH" value around, essentially channging the zero point; so our pressure gauge field depends on how we measure, as well (different observers will report different values at the same point!). Moreover, very fast moving pressure altimeters (naively) may report different values compared to those that are slowly moving low in the subsonic regime. However, by and large, if two airplanes are near one another with their pressure altimeters corresponding to FL10, they have to be careful not to collide! So our pressure gauge field does have its uses in predicting physical events like collisions, even if there's nothing truly special, universal or even locally constant about FL10 or any similar surface.
Note that when we slide around zero points, the physical differences don't change. You're still suffering fever symptoms whether we tell you your temperature in degrees Rankine or kelvins-above-normal-body-temperature. The time in seconds from one northern winter solstice to the next is essentially the same whether you use the Gregorian, Julian, Chinese, or any other calendar to fix the day, month and year.
When we fix these zero positions and label the distances from them in degrees or years or volts or whatever, we are essentially laying down a set of coordinates that are at least locally valid. This lets us concentrate on local interactions without carrying around additional burdens from considering how how our local zero and gauge-markings might not be appropriate elsewhere in spacetime (or to a different culture, or whatever), and generally we can convert from one such fixing to another.
Although it is tempting in the pressure case to say that the zero is universal and so only the choice of units (pascals, pounds per square inch, whatever) is idiosyncratic, let's think about how to represent pressure without making any such choices.
In General Relativity pressure is encoded in the stress-energy tensor as the doubly-spatial components (T_{11} = T_{22} = T_{33} = p) where 1 2 and 3 are the three space-like axes and p is the total pressure. T_{12} is the flux of 1-momentum in the 2 direction, T_{13} is the flux of 1-momentum in the 3 direction, so T_{11} is the flux of 1-momentum in the 1 direction. If we make the 1 axis "left" and "right", then we can think of T_11 as "the flux of momentum from the left going towards the right through a point". That's just pressure pushing on something to the right (or a tension pulling on something to the left!).
We can assign any sort of label to the 1 2 and 3 directions as long as they're orthogonal. If 1 is left-right then 2 could be forward-backward and 3 could be above-below. Or we can say 1 is x-axis, 2 is y-axis, 3 is z-axis in Cartesian coordinates. Or we can say 1 is radial distance, 2 is azimuthal angle and 3 is polar angle in spherical coordinates. And so on. We can also set down units, whether they're metres or feet or light-seconds or anything. And finally, we can set down some origin for this system of coordinates. Whatever such choices we make, we quantify the relative momentum flux, but we don't actually change it.
That there is a flux at all is invariant in the face of these choices of coordinate and gauge. (The more precise statement is that the divergence of the total non-gravitational stress-energy vanishes, so one could in principle concoct an observer that sees the pressure-tension as some other form of stress-energy).
But your question is essentially, "what if there is nothing to the right to push on"? Well, if there is a flux of left-right momentum in the right direction, but nothing to the right, then for a truly pointlike object, there's now less stress-energy in that object and a bit of stress-energy to its right. Conversely, there is now some stress-energy to the right of the pointlike particle that has our test particle as a neighbour to its left. Depending on the gauge theory of the particle, we can now consider what becomes of the two neighbours: do they drift apart? Do they rejoin? In the former case, our gauge field can carry away the momentum to infinity; in fact, that's exactly what photons do: they are gauge bosons, massless, but not momentum-less. In the latter case, our gauge field might pull the momentum back inside the particle, essentially how asymptotic freedom works for objects that feel the strong nuclear force. And indeed the strong nuclear force is again representable as a gauge field: the gluon (massless, but not momentum-free). Photon pressure is even testable with e.g. solar sails. Gluon tension (tension being the inverse of pressure) is seen in hadronization.
One molecule of nitrogen is surprisingly busy with tension and pressure being carried around locally by gauge bosons!
Finally, depending on how deep you mean by "dig deeper into the math", you (or someone else reading this, including me at some point in the future :-) ) might like Terry Tao's take on this sort of thing: https://terrytao.wordpress.com/2008/09/27/what-is-a-gauge/ If nothing else, his approach to "explain[ing] what a gauge theory [is]" there is a bit different from mine here. :-)
There's a whole set of excellent reasons to prefer to work in inertial frames where possible, most of them having to do with calculational burden. Our case here is essentially flat electrovac with a test field - do you really want to work in Rindler or Born coordinates or worse if you don't have to? You're free to do so if you're feeling masochistic (or doing it as a learning exercise), just as you're free go further and solve Maxwell's equations in curved spacetime in arbitrary coordinates with the vector potentials in some arbitrary gauge (\partial_{\mu} A^{\nu} \neq 0). And with "proton?" then you might as well dive into QCD as well. Unnecessary extra work is surely a good reason to prefer to use the simpler maths where the results will be indistinguishable.
> Without the laws of motion
I don't understand this point. Are you ditching the action principle somehow?
There are no meaningful potentials in the one-particle universe that the OP described. However the particle is "really moving," you could just invent a coordinate system to make it "move" however you like.
Even introducing EM would bump you up to at least two particles! The only reasonable one particle universe would be one where the partice was coupled to nothing.
(It is under the "try to interpret it as they meant it" doctrine that I go from a proton to a particle that doesn't leave behind a disturbed field when it accelerates.)