For instance, what does an observer hovering just above the horizon of a different black hole see?
There is more than one astrophysical black hole in our universe, and one should not forget that when reasoning based on the Schwarzschild exact solution. Worse, most of the astrophysical black hole candidates have non-negligible spin. Certainly it would be weird for a stellar black hole to have low spin, as they will have previously have been rotating stars, and so with any sort of trust in the exterior Kerr solution[1], it seems unwise to ignore the observables generated by the ergosphere, in particular the non-stationarity of objects within it.
It doesn't really change the thrust of my question two paragraphs up: can an accelerated observer see a blueshifted infall more similar (in terms of counting time by the ticks of the observer's wristwatch) to the left than to the right in https://en.wikipedia.org/wiki/File:Gravitational_time_dilati... ? Or, equivalently, are there families of observers who are not (in this case Boyer-Lindquist) coordinate observers ? (The second answer should be, (a) yes, we can find arbitrarily accelerated observers, and (b) we may not be able to find a mapping among the local inertial frames of the infaller, the coordinate observers, and arbitrarily accelerated observers, but we should not ascribe any physical meaning to our failure to do so. cf [2]).
Perhaps a more interesting way of thinking about it is that there is essentially no practical difference between a black hole with the infaller collided with the gravitational singularity and a black hole with the infaller at 2GM+epsilon (in units such that c=1 etc) above the gravitational singularity, and if there was a practical difference and it persisted longer than a light-crossing time, we would have disproven the no-hair conjecture. However, LIGO/VIRGO evidence shows a very clear ring-down for BH/BH and BH/NS mergers, which fails to support such a challenge to no-hair.
For us weakly accelerated Earthbound observers, neutron stars and black holes at a wide variety of distances from us completely fall into each other in finite (indeed, short in human terms) time.
Non-compact infallers don't raise much of a bump on the horizon of the black hole in comparison, but the result is the same: M changes (as does the spin parameter a), and we now "find" the horizon at a new set of points. (Here it's tempting to talk about slicing up spacetime into space and time so we can talk how it is easy to forget what we mean when we talk about e.g. the "before" horizon and the "after" horizon, or alternatively discuss whether, if we had a sensitive gravitational wave detector, we could use that to decide whether a large-mass infalling object was inside or outside the black hole).
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[1] There are objects like https://carinaemajoris.wordpress.com/2012/07/07/a-very-errat... whose observations are much closer matches to Kerr (exterior) solutions with spin parameter ~ 0.9 than to Kerr (exterior) solutions with spin parameter ~ 0 (in particular they're pretty clearly not generating the sorts of timelike geodesics that one would find in a Schwarzschild spacetime).
1) When a star collapses, the black hole it becomes isn't formed instantly, that would violate the very theory of relativity that predicts their formation.
2) Black holes evaporate, as you've mentioned.
But this begs the question: Forget the hypothetical "probe" particle. What about the particles that made up the original star? They're like many probe particles!
Imagine the black hole starting at the centre of the collapsing star, expanding outwards, slowing down time for the infalling matter. Does any of the rest of the star fall in to the black hole? Or, like your test particle, does the rest of the star just hang there, frozen in time, a thin layer above the event horizon?
Regressing that back to the formation: Does anything ever truly fall into the black hole?
I suspect not. My thinking is that a black hole truly is a hole in spacetime, and contains nothing, not even spacetime. It's like poking your finger through a loose-knit wool jumper and making a hole. From the perspective of the threads, you've made a distortion that can be approached, but not entered.
Then the evaporation of the black hole is the hole in spacetime closing up, releasing the compressed matter near the horizon as Hawking radiation...