1. Yes.
2. In that case, Player 2 will get the mate, but in certain scenarios, if the king's flight squares are removed by a move that doesn't actually put the king in check then the player to do that will get the mate.
3. The person who moved the piece should get the mate. I have yet to verify that in a game though.
So for #2, if I understand you correctly, if Player 1 puts Player 3 in check and Player 2 blocks his escape routes, Player 2 gets the mate.
A couple more questions:
1. What happens if Player 1 checkmates Player 3, and then Player 2 blocks the check or creates an escape route? Is Player 3 still alive? Does the checkmate process as soon as it's put in place, or not until its Player 3's turn?
2. If Player 3 is still alive in that scenario, what happens if Player 2 doesn't interfere with the checkmate, but instead puts Player 1 in check (or checkmate) on their turn? Are Player 3 and Player 1 both eliminated, with 15 points each for Player 1 and Player 2?
Checkmate happens as soon as the move is played and player 3 will be out of the game.
1. What happens if Player 1 moves so that Player 3 is now in discovered check from Player 2? Can Player 2 take Player 3's king and get the points, and Player 3 is just dead?
2. What happens if Player 1 uses his rook to pin Player 3's king on the back rank, and Player 2 brings his rook down to do the two-rook checkmate? Who gets the points? Player 2, since he's the one who actually put the king in check?
3. What happens if Player 2 moves a piece so that Player 3 is now in discovered checkmate from a Player 1 piece? Who gets the points? Player 1 for being the one threatening the king, or Player 2 for making the move that resulted in the checkmate?
These sorts of questions are part of the reason we always allowed leaving yourself in check and moving into check in our three-handed chess game, and someone had to actually capture the king to kill you. That's not the only way to solve the problem, but it's an easy way. And in our three-handed version where you took control of the pieces of the person you killed, it sometimes resulted in interesting strategies when people would intentionally leave themselves in check to force another player to bail them out; that strategy wouldn't usually work as well in a points-based system, though.