Probably not because the 2 kings can't get closer than 1 square away from each other.
It would be refereed to as check, but if these are the only two pieces both would be in check, so whom ever would have the next move could technically move his king out of check since there would most likely be open spaces since the king is highly protected at the start and can move one space in any direction.
No. In order for Black King to capture White King, it would have to get within one square of White King, but in order for to that happened, Black King would have to place himself in check and that is against the rules of chess. So if that situation occurs, a stalemate would result and the game would end in a Draw(tie).
I think no, oh wait maybe.... But I'm not sure with my answer.