This is a drawn game according to the rules of chess, since checkmate is impossible.

Other draw conditions include:

* Stalemate (=The side to move has no legal moves)

* 50 moves in a row without any permanent changes happening ("permanent changes" means a pawn moving or a piece being captured)

* Exactly the same game state occurs for the third time ("game state" is whose turn it is + status of castling ability + status of en passant possiblity + which pieces are where)

* Agreed draw

Fun fact: In normal time controls (not blitz) at competitive levels, chess games virtually never end in one of the termination conditions. I don't have statistics on hand but I would expect resignations and agreed draws account for 98%+ of grandmaster-level games.

As I understand it, the threefold repetition rule does not mandate a draw, it just allows either player to claim a draw upon occurrence.

I believe 50 moves rules is the same. But in that case, one player is more interested in the draw than the other. In the case of repetition, both players are avoiding a move that gives the other the advantage. It's not so much a mandatory or not question, but if one of them is more willing to try a different move than to draw.

That's right, see article 5.2.d under the FIDE Laws of Chess:

http://www.fide.com/component/handbook/?id=171&view=article

Yes, its because no other piece can take a king but a king can take all other pieces. The case for taking a king is actually quite rare!