Could a ship equipped with Alcubierre drive theoretically escape from a black hole?
Also, could it reach parts of the universe that are receding faster than the speed of light from us?
Theoretically, yes. Assuming that our physics for black holes is correct enough to tell us that the escape velocity of a black hole is just greater than the speed of light. If you think about going faster than the speed of light as going back in time, this also serves as a way of "escaping" the black hole, for you would go back in time to a point to where you were never in the black hole. Therefore, you "escaped."
If Alcubierre warp bubbles are physically possible, which is exceedingly unlikely, and if the equivalence principle is correct, you could definitely escape from a black hole in one, because there's nothing locally special about the event horizon. In a large black hole the whole escape process can happen arbitrarily far from the singularity, in a spacetime region that's arbitrarily close to flat (aside from the Alcubierre craziness). However, if Alcubierre bubbles were possible then it might be more accurate to say that there is no event horizon and no such thing as a black hole interior in the first place. Some people working in quantum gravity think (for totally different reasons) that GR's model of the interior is totally wrong and you would actually disintegrate upon hitting the location of the apparent event horizon. In that case you couldn't escape, there being nowhere to escape from.
Reaching very distant parts of the universe is not so easy, because the outside of the Alcubierre warp bubble, where some of the exotic matter is, is spacelike. Unless the exotic matter can move locally faster than light (which would take away a lot of the interest of the Alcubierre solution), you have to start preparing the Alcubierre geometry from a spacetime point whose future light cone includes the entire trip. (See Alcubierre drive#Placement of matter on Wikipedia.) So if you want to travel a billion light years away, you have to start preparing for the trip a billion years before it starts. This doesn't make the whole thing useless, since you can start preparing the return trip at the same time and make the round-trip time arbitrarily small (or negative, for that matter). But it would preclude reaching anything outside the future light cone of whenever we solved all of the other problems, unless Alcubierre bubbles occur naturally or obliging aliens did the setup for us. In a Lambda-CDM universe, that means that we couldn't reach some of the galaxies that we can see, since they will never intersect our future light cone.
The drive works by warping normal space, creating a bubble that kind of surfs through space time. I don't know what would determine the speed such a ship could achieve so not sure if a natural law would limit the ability to travel beyond visible space. Black holes exist because their extreme mass has warped space beyond to point where light can escape. It seems like it might take an infinite amount of energy in this case to keep the warp bubble from collapsing.
In my understanding, which is based on simple facts and admittedly not a big understanding of how black holes would work with regard to this...
A warp bubble would compress space in front of the ship and expand space behind the ship. This should effectively move the black hole away from the ship, descreasing the effort needed to escape. It also means that if you had fallen past the event horizon, you might find yourself outside it again as the warping of space moves the black hole, and thus the event horizon away. At this point, I suppose it would come to this: is the speed of the warp bubble a factor in your escape velocity, or does the bubble only serve to buy you more time, and you need to have a velocity within the bubble to actually escape? I'm not the one to answer that lol