It will gain energy.
There _is_ no amount of energy it could gain to escape though. It goes no faster, gets no closer to escape. It just splats into the black hole and we'll never see it again no matter how much energy it started with or gained.
It's not a contradiction because the path of the light enters the black hole event horizon but does not come back out.
All paths end at the ~center of the black hole. We don't really know what's there, at the so-called singularity at the center, it may be impossible to know. But it's probably something that light can hit and stop existing, or even if it can't it'll be stuck there in some way or another.
We know that the gravity (warping of spacetime) around a black hole is such that it cannot escape, even at light speed. What happens inside is less sure, but largely moot because none of it can ever affect anything outside of the event horizon ever again.
(Hawking Radiation or something else may make this technically a lie, but it's close enough to the truth to last a few trillion years)
Maybe. Nobody really knows what happens at the center. General relativity I believe says "yes", but nobody knows how true it is in the case of ~infinitely dense crap in an infinitely small spot. Usually "infinite" means what really happens is not captured by the model, but nobody has a better answer yet as far as I know.
A better answer would be something like a grand unified theory of gravity and the other forces, including quantum effects, which doesn't exist. It's like the holy grail of physics, you'd be the next Einstein if you figure it out.
From an outside observer, the black hole is finite in width though, though. and if the speed of light is constant, isnt that a contradiction? Or is a black hole somehow of infinite width?