Giant body of water found in space, black hole claims it was just hydrating

As for the weird distances, just imagine an ant moving across a rubber surface at constant velocity. The rubber is slowly stretched over time, so while the ant continues at the same rate it will actually cover less space. If the rubber continues to stretch, there is an event horizon effect, meaning some distant parts of the rubber sheet will become inaccessible.

The inflationary epoch is rarely understood, but it's really so qualitatively different from everything that came after that it's worth referencing as a starting point for the Universe as we know it. Intuitively, you can think of the inflationary epoch as an extremely brief flash of time where things were so bizarre that nobody really has any firm idea what the hell happened. What we do know is that somehow the Universe expanded by a linear factor of at least 10^26 (it could actually be septillions of times higher; we just have a lower bound) within a tiny, tiny fraction of a second and then pretty much stopped.

Theoretical physicists mostly agree that the Big Bang happened a tiny fraction of a second before the inflationary epoch ended, but it's all speculation at that point. Pragmatically, it's very useful to call the end of this epoch the "beginning" of the Universe even if strictly speaking it started a few trillionths of a second earlier. Inflation was so fast that you can't really describe anything useful in terms of pre-inflationary distance. Everything that we could ever observe or care about in the Universe all emerged from some volume of space that was smaller than an atom until right before inflation stopped.

Why inflation stopped is very much beyond my understanding, but the metric expansion process that took over afterward, Hubble expansion, is negligibly slow by comparison. It's so slow that it's generally ignored in any context with time scales shorter than hundreds of millions of years. This is an important point for establishing what exactly we're talking about when we say the "beginning of the Universe". As far as the expansion of the Universe goes, the places that things were in 10 seconds after the Big Bang is more-or-less where they were the next day, and the next year, and the next millenium. Hubble expansion only matters over really big distances or really long times, like billions of years.

So, to finally answer your question, this is why I just said "in transit since the start of the Universe" without any other explanation. In practice, it doesn't matter whether you take it to mean "10 seconds after the Big Bang" or "10,000 years after the Big Bang", just so long as you're referring to distances that applied somewhere near the start. Most notably, the Cosmic Microwave Background Radiation was emitted by an event that occurred only 300,000 years after the Big Bang, and there's good reason to think we're not likely to ever do better than that in terms of what radiation we can feasibly detect. This is why you hear it described as "remnants of the Big Bang", though really it was more directly caused by the formation of hydrogen 300K years later.

The 46 Glyr figure is admittedly ambiguous, and could refer to either of two things. The first is what's known as the surface of last scattering. This is what the CMBR is all about. The CMBR basically happened everywhere at once, and as you go further into the future the light that you're seeing from it came from somewhere that was further and further away at the time the CMBR was emitted. Right now, the light we're seeing from the CMBR came from points in space that are currently 45.7 Glyr away. (In cosmology, distances are almost always given in terms of present-day distance for consistency. This is called the "co-moving" distance if you want to be explicit, which alleviates confusion that might result from just calling it the "current" distance.)

The second thing the 46 Glyr could be is what I was actually referring to: the radius of Earth's particle horizon. This is the co-moving distance between us and the most distant point that would be reached if you emitted a light-speed signal from Earth's location right after the end of the inflationary epoch. This also coincides with the "observable universe", the portion of the Universe that was sufficiently close to us at the end of inflation that signals emitted from it could in principle reach us today. The two concepts are really equivalent, just with different frames of reference. This region has a boundary that's about 2% larger in diameter than the surface of last scattering. That extra 2% corresponds to the distance that radiation could have traversed theoretically (but not actually, due to plasma interference) during the 300,000 years that elapsed before the CMBR was emitted.

I don't have time to give a proper answer to your "at rest" question, but very tersely: The assumptions about simultaneity and privileged reference frames that get pulled out from under you by special relativity are actually pretty reliable in cosmology. You can, as a practical matter, be very sure whether or not your space ship is moving or not by looking at the stars around you. Most people are so caught up in the idea that the Earth is moving around the Sun and the Sun is moving around the Milky Way that they never bother to learn that the pattern doesn't actually go any further. The largest-scale structures of space are foam-like clusters of billions of galaxies, which become less distinct and more homogenous as you zoom out. Galaxies themselves do very little movement beyond drifting along with the metric expansion of the space time they're sitting in. Most "peculiar movement" is caused by gravitation within galaxies, not between them. This is quite literally what a galaxy is: It's a cluster of matter where gravity causes stars to have a peculiar velocity that cancels out the metric expansion of the space between them.

Further reading: http://arxiv.org/abs/astro-ph/0310808