Regarding the castling question, I would have thought that the logical generalization of the orthodox chess rules would be to allow castling out of or through check, but also to allow the King to be captured en passant on either its original square or the square it passed through. This would of course result in a loss unless the castling player had a second King on the board.
Also, I've just thought of the following pathological possibility. Suppose that, after 41 moves of a game of Extinction Chess, White's only Pawn is on b7 and Black's only Bishop is on c8. If White then plays 42. bxc8=Q, Black's Bishops are extinct, but so are White's Pawns. So the game is clearly over, but what is the result?
Also, I've just thought of the following pathological possibility. Suppose that, after 41 moves of a game of Extinction Chess, White's only Pawn is on b7 and Black's only Bishop is on c8. If White then plays 42. bxc8=Q, Black's Bishops are extinct, but so are White's Pawns. So the game is clearly over, but what is the result?