Could a ship equipped with Alcubierre drive theoretically escape from a black hole?

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 in advance. 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.

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."