> from inside the black hole, it must look like everything that has ever fallen into the black hole in the history of the universe has fallen into it at the same instant.No, that's not correct. You can see things that have fallen in before you, but not after you. There are still distinct points on the horizon where things can cross, and distinct trajectories inside the hole.
> I'm not entirely sure how black hole evaporation fits into this
Nobody knows for sure because we don't have a good theory of quantum gravity. In Hawking's original semi-classical model of black hole evaporation, things that fall into the hole before it evaporates are still destroyed at the singularity, but if you wait outside and watch the hole evaporate, the final burst of light as the hole finishes evaporating and disappears will contain images of everything that fell into the hole, at the instant it crossed the horizon. However, it's not at all clear that that model will still be a good approximation when we have a full quantum gravity theory.