Can things move faster than light inside the event horizon of a black hole?

I think your initial question is a good one, but the text gets a bit more jumbled and covers a few different points.

Can things move faster than light inside the event horizon of a black
hole?

Nice question.

Black Holes are regions of space where things get weird.

I'm 100% OK with this statement. I think it's a true enough summary and I'm sure I've heard physicists say this too. Even if "Weird" isn't a clearly defined scientific term, I'm 100% fine with this (even without a citation).

Past the event horizon of a black hole, any moving particle
instantaneously experiences a gravitational acceleration towards the
black hole that will cancel out it's current velocity, even light.
That means that the gravity well of the black hole must be able to
accelerate from -C* to 0 instantly✝.

Are you quoting somebody here? Anyway, this isn't quite true. Black holes don't accelerate things from -c (which I'm guessing would be a light beam trying to fly away from the singularity but inside the event horizon), to 0 "instantaneously".

Perhaps a better way to look at it is to consider curvature of space, and inside a black hole, space curves so much that all directions point to the singularity. It's the "all roads go to Rome" scenario, even if you do a complete 180, you're still on a road that leads to the singularity.

I understand the temptation to look at that as deceleration, but I think that's a bad way to think about it. Light doesn't decelerate, it follows the curvature of space.

Given that fact, we can assume the gravitational acceleration of black
holes is C/instant**. Given this, it stands to reason that in
successive instants, the particle will be moving at speeds greater
than C, because it is experiencing greater gravitational forces and
continuous gravitational acceleration.
Does this actually make sense? Is there something I'm missing here? By
this logic, it seems like anything inside of the event horizon of a
black hole could and should move faster than C due to gravitational
acceleration.

outside of a black hole, continuous acceleration would never lead to a speed greater than C. You can accelerate for billions and trillions of years and all you'd do is just add more 9s to the right of the decimal point.

You seem to be assuming that inside a black hole this can happen, but I'm not sure why you'd assume that.

"continuous gravitational acceleration" - no matter how strong, is no guarantee for faster than light travel. That's logically inconsistent with the laws of relativity.

Edit: I showed this question to a friend and he questioned if the
hypothetical particles that were radiating from the singularity (The
photon traveling exactly away from the black hole) might be hawking
radiation; that is, the gravitation acceleration of a black hole is
only strong enough to curve the path of light around a non-zero radius
(thus not actually stopping it, but altering it's course), and not
powerful enough to decelerate light. Is this actually what hawking
radiation is, or is he as confused as I am?

I think, a more correct way to look at hawking radiation is to see it as something that forms just outside of the black hole, a particle/anti particle pair and one escapes and the other falls inside, and that's probably not 100% correct either, but the singularity itself doesn't send out particles. Hawking radiation has to do with quantum properties of space. It's not a property of black holes. The black hole just happens to be unique in that it can capture one half of a virtual particle pair and the other half can escape.

This also is a pretty different topic than your original question.

*Where movement towards the singularity would be considered a positive value, movement away from the singularity is a negative value, that
is, anything moving at the speed of light away from the singularity
would be moving with a velocity of -C relative to the singularity.
✝If it couldn't accelerate from -C to 0 instantly, any photon
traveling exactly away from the black hole would be able to escape the
event horizon.

**An instant is an arbitrary amount of time, it could be a fraction of a second, a second, a minute....

I think it's a good idea to differentiate mass-less pure energy particles and particles with mass. You seem to be saying that a ray of light can be traveling away from a black hole at the speed of light, get caught in the gravity, slow down and then fall back into the black hole like a ball that's tossed straight up into the air from the surface of the Earth. That's probably not what happens. The ray of light follows the path of space time ahead of it, which happens to be curved so much that it points into the black hole, even if, in the classical sense, the light begins by pointing away. All space curves into the singularity once you're inside the event horizon, so there is no "away from" anymore. At least, that's how I think it works.