UCLA GCG and other efforts have failed to image occlusion of orbiting stars or gravitational lensing effects while viewing Sagittarius A* (the supermassive black hole at the center of our galaxy), thus it may not actually be a black hole (instead, it may be even stranger, like an absolutely gigantic plasmoid that somehow ties the room together).
I know they meant theoretically poke it, but we've been having trouble finding nearby ones.
As qubex suggests, this is really about confirming that asymptotically Schwarzschild spacetime (and its close relatives) is mathematically robust [1]. Asymptotically here means that if we perturb (gravitationally or electromagnetically) a Schwarzschild black hole the resulting radiation from "balding" will decay exponentially fast and die off to ground state well before spatial infinity. This is also a useful check on practices like (in cosmology) taking a boundary at some reasonable radius from a galaxy as if it were asymptotically Schwarzschild space, and stitching it into the expanding Robertson-Walker spacetime.
There is no poking of astrophysical black hole candidates involved.
That said, signals from LIGO and (eventually) LISA will provide experimental evidence to check some "balding" conjectures for various types of extremely strong perturbations (NSes falling into BHs, for example). Electromagnetic astronomy isn't really directly helpful, other than double-checking that one of the colliding compact objects is almost certainly not a black hole.
Finally, "other efforts" will soon include the Event Horizon Telescope, which was already the only practical platform for direct observation of a candidate BH. For Sgr Astar, observations from the South Pole Telescope have to be captured in Antarctic autumn and this time around the recorded data had to wait in Antarctica until Antarctic summer (when the physical media could be flown out in reasonable weather; there is insufficient bandwidth from Antarctica to send it over a network), so the first tranche of data are still being crunched. It's very premature to do wild speculation about the nature of Sgr Astar (and your speculation is really wild).
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[1] Hintz works on the stability of several black hole solutions, for example https://arxiv.org/abs/1612.04489 -- he here he deliberately breaks the symmetries of the Schwarzschild solution to see if removing tiny angular momentum by hand poses mathematical problems.
I haven't looked at the Klainerman and Szeftel paper [ https://arxiv.org/abs/1711.07597 ] yet.
You can have a very mild ("no drama" conjecture) black hole event horizon for ultra-massive black holes. The curvature at the horizon gets gentler with higher-mass, so you can envisage an indistinguishability of the tidal/shear stresses at the horizon versus the tidal/shear stresses in empty space many many light years outside the horizon. You still won't get your hands back.
The horizon isn't actually any sort of surface (nothing bounces off it) but you should sure feel the lack of nerve impulses returning from whatever bits of your anatomy are on the inside if your brain is on the outside. Ouch. Also, the horizon is likely to be sufficiently "sharp" that on lots of reasonable orbits above it that let you dip your hands in (without all of you falling in too), your hands will be fairly neatly and very rapidly torn off. (It would be fairly easy to misjudge the exact location of a horizon though.)
I suppose if you have a sufficiently small black hole (and ignore the extreme hostility of the environment near it) then if you have lots of "shielding" it is plausible that you can move the location of the horizon (from your perspective) such that your hand can go to where the horizon would be without the shielding. I don't think that would really count as "touching", though. It's a little like blowing on the surface of a soap bubble then waving a pin around where the film was (and returns to) in the absence of the blowing air. "Look, I didn't pop it!" It's not like putting a bit of scotch tape on a balloon and putting a pin into the tape ("look, no pop!").
Would your hand even be connected to you at that point? The bonds holding the molecules together wouldn't work, right?
I won't try sketch a semiclassical gravity attack on the second question, which is the harder of the two, and for the first, I'll mainly feint and point you to Greg Egan at http://www.gregegan.net/SCIENCE/Rindler/RindlerHorizon.html as a starting point.
On the one hand, for a sufficiently massive black hole, the Rindler solution he explores is an excellent approximation. On the other hand, his exact solutions are (a) not fully applicable in the dynamical spacetime of a sufficiently massive black hole (even one with lots of symmetries and otherwise in isolation) because the Rindler horizon is a local structure while the BH horizon is a global one [1]; there is a local boundary that forms a point of no return for objects near enough a black hole though you have to calculate where that is; (b) Egan's rope is a classical object, whereas when we're at the level of cellular signal transduction, molecular bonds, and chemical bonds, classical simplifications are already probably cheats or at least misleading. On the other hand, there are lots and lots of particles involved, and tracing the evolution of each of them in some suitable coordinates would be an enormous amount of work.
So I trust my own intuition only to the extent that (as Egan notes) there are some decent mathematical similarities between the Rindler horizon and a BH trapping surface.
Sadly, there will be no contact with observation in our lifetimes, but we might make quick and dirty numerical solutions that will give a strong theoretical prediction. Additionally, it is at least plausible that we will be able to do small-scale tests of Rindler space in a few decades. Until then be wary of people offering glib responses, especially really wrong glib responses like "the hands will never pass through the horizon because of time dilation" etc., which are sadly commonplace in online forums.
Finally, my own thinking was that a powered hyperbolic orbiter (rocketman!) with sufficient momentum dropping his hands past the local point of no return would end up with stumps, and what does the actual ripping is rocketman's momentum [2]. In my head is a picture of Wile E Coyote running into a quicksand (or cement or tar etc) trap and either being tripped up and pulled into the quicksand or being unlucky enough to have enough forward momentum that he leaves his feet behind on the first step. Roadrunner corretctly judged exactly where the local trapping surface was and so skimmed right over the trap; Coyote miscalculated where Roadrunner's trapping surface would be (given more global knowledge of the configuration) and then his own.
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[1] In particular, BHs (i.e., compact masses sourcing a trapping surface) can last extremely long times compared to the age of the universe and can in principle be observed by anyone in the BH's Hubble Volume, while real objects cannot be Rindler observers for very long (what fuels the acceleration?), and the Rindler horizon is peculiar to the Rindler observer rather than a trapping surface. Nevertheless, one can take the analogy seriously [ https://arxiv.org/abs/1305.4986 for instance ]
[2] So I can "cheat" here by having the ripping happen between bits of tissue that are all outside the horizon at the time of rip; the tissue that is already inside the horizon is then irrelevant, and it's mainly whole cells that fall in. The centres of momentum and mass of the unfortunate astronaut are highly dynamical during this, so it's not so shocking a cheat.
The gradient is much lower for a much larger black hole, so this sort of drag becomes apparent ever closer -- or even right at -- the outside of the horizon. For a sufficiently large black hole, dipping one's hands towards the black hole will probably not feel any different compared to deep space, even very close to the horizon. However, one would then expect a short sharp shock dipping one's hand into the horizon. Bye-bye hand.
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[1] Put the rocket pack on the closer-in side, and the intermolecular forces etc that hold the astronaut into familiar shape will decelerate the whole astronaut along a suitable choice of coordinates. The coordinates you want are probably integral over Fermi normal ones for ever-smaller components, lots of fun to calculate.
Its diameter is smaller than the Sun-Mercury orbit[1], plus the stars in close orbit have a period of 11 years at the shortest. So they're far away from A* plus they don't transit very often.