wetware: As for (1), what I read in the Encyclopedia of Chess Variants is that Fred Galvin issued two games in 1958, Refusal Chess and Compromise Chess. Compromise Chess is played with the rules you stated, and Refusal Chess is designed as the over-the-board version, where you have to play one of your two moves over the board, and the opponent can reject it if he wants. It is virtually the same game, but for a reason I don't know in Refusal Chess Galvin decided for the take-the-king version.

Now, I am not saying that one rule is better than the other. I generally prefer take-the-king rules, but in this case it is not a real take-the-king rule, because if it was, the mated opponent could refuse the move which takes the king, so a double check would be needed to win the game (duh!).
Regarding endgames, I don't think that it should not make a big difference (but I share your bias!)

mangue : Where did you read that ? That must be a typo :-)