Is this considered checkmate if I touch a piece which can't make a legal move? What if there's a touch move rule, and my knight or bishop is blocking my king, and I touch it. Is that a checkmate? If not, what is it called?
No. The touch-move rule can't force you to make an illegal move, and it is not legal to put yourself in check.

Here's what the FIDE laws of chess say:


  4.5 If none of the pieces touched in accordance with Article 4.3 or
  Article 4.4 can be moved or captured, the player may make any legal
  move