Anti-matter would increase the mass of the black hole, not distroy it. I don't see any way that quantum teleportation could be useful here, if that's what you meant. There is however a similar and simpler idea called the Alcubierre warp drive. Unfortunately it requires that we are able to produce large quanties and control a type of matter that has never been detected. Even if such matter exists, it would almost certainly be absurdly difficult to produce it and control it, and even if we could, it would be disappointing. Nothing at all like Star Trek.
No. What we can do is to transport the quantum state of a single photon. What that process does is essentially to change the polarization of a photon that's already at the "destination" to what was previously the polarization state of a photon at the point of "departure".
Yes, in 1932. (That's when the positron was discovered).
As I said above, it can't. I don't think anything can, except time. Just wait much longer than the current age of the universe, and it will destroy itself by emitting Hawking radiation until it's gone (or whatever happens when only a really tiny piece is left...no one really knows).