The resolution of these telescopes is limited by diffraction, not by the number of pixels on the sensors. The achievable angular resolution is roughly the wavelength divided by the aperture diameter [1]. JWST works in the few µm wavelength range and has a 6.5 m aperture, such that the angular resolution is ~0.1 arcsec. The EHT works with 1.3 mm wavelength and has an effective aperture of roughly the earth diameter (~13000 km). This leads to an angular resolution of a few ten µarcsec which is more than 1000 times higher than that of JWST.
The edge should be quite sharp. Any deformation or movement in the edge will be smoothed extremely quickly, on timescales comparable to the light-crossing time of the object -- in this case ten seconds or so.
If you're a photon and you're in, you stay in. If you're out and heading out, you get out. (if you skim the surface, you might make an orbit and then leave :) ). It is that fact that makes the edge quite sharp.
I disagree. The edge of a BH is essentially an asymptote. While there is a mathematical bright line, when looking at it you should see light in all manner of red/blue-shifted colors near the event horizon. Since that light is coming in from a variety of directions it leaves in a variety of directions too. Everything would look soft and fuzzy around the edges. Out of focus.
Out of focus? I don't know about you but I find the black hole "edge" in the simulated images in Interstellar quite sharp.
(Or are you talking about resolving matter/light near the event horizon? In that case I agree – one won't really resolve any structures anymore due to light getting bent and redshifted in a myriad of ways.)
In case you weren't being rhetorical, the sun's edge would either be the chromosphere[0] or the corona[1], the corona being famously fuzzy and also quite bafflingly hot. The chromosphere is "smoother", but still very interesting on any given day [2].
this is sort of getting into the definition of black holes and event horizons. I don't think they really have solid surfaces, I would expect all imaging here to show fuzzy samples.
I couldn't really say for sure but I think macroscopically (viewed from a low-resolution telescope) it would look fuzzy, but close up, it would look very spiky and dynamic with all sorts of stochastic events happening.
I think you just sort of rephrased the question :) Also, if you have a photo of a tree, then there are infinitely many objects that will produce that photo; however, that doesn't make it a bad or worthless photo.
But I suppose you could be right for a single image from one angle, and I suppose that we don't get to see this particular object from many different angles.
[1] https://en.wikipedia.org/wiki/Angular_resolution