It’s called nostalgia. 
While it’s speculative VSauce on YouTube does a pretty good job of describing a trip into a black hole and what you would see: https://www.youtube.com/watch?v=3pAnRKD4raY
If I remember right, ours will poop out sometime after a series of helium flashes and settle down into a white dwarf, without actually fusing the carbon or oxygen that built up in the core.
Our Sun sucks. Of course it’s an underachiever.
That was neat.
Suppose I’m above a non-rotating black hole, and I’m looking away from it. I drop towards it, looking away from it until I cross the horizon. What direction am I now looking?
The video linked above answers that question somewhat. The circle of light coming from outside gets smaller and smaller and vanishes into a point. I’m a little unsure exactly when it vanishes completely, though.
It seems to me there are two different things we might be talking about here. One is what we SEE, which is based on the incoming light. The other regards a direction we might go. Evidently, light can appear to come in from a direction in which we can’t possibly go. Or something like that.
Yes, both those are coordinate independent properties. That said the first statement is true for any future light cone, so the defining property is that what’s inside the event horizon doesn’t lie in the past of future null infinity. The second statement would be true I believe only if certain energy and causality conditions are obeyed.
You can still see light from objects outside the event horizon (though obviously what you see is severely distorted) and you can still move in the direction from which the light is coming from. One way to imagine it is that the event horizon is now receding from you faster than the speed of light, so for an observer in the black hole even increasing their velocity towards the horizon won’t stop it from getting further away.
Analogous to racing to the edge of the visible universe or is there a difference?
AFAIK, light from an object falling into a black hole becomes infinitely redshifted on the way out.
I suppose, from the singularity’s point of view, all light on the way in is infinitely blueshifted.
Does the influx of ultra-high energy make the growth of a black hole greater than simply absorbing non-blueshifted energy?
It would be infinitely blueshifted from the point of view of someone at rest within the hole… except that there is no such point of view. It will not be infinite from the point of view of an infalling observer.