Is my engine drunk, or is there a good reason to move the King here? 

Here my engine recommends 11. Kf1, and scores the position after this move at +2.3. After 11. Bb2, the engine gives a score of just +0.9. Why in the world is it better to move the King, as lose the right to castle? Is there a good reason, or is my engine drunk?

There are som positions where engines have no clue. I don't know if this is one of them, because Kf1 seems to have it's merits. It gets out of a potential pin on the king, but other than that I don't know. I can't believe that white is up the equivalent of two pawns in this position, or is anywhere near ariving at a position where he is.

I agree. It's wicked. 

11. Bd2 Bxd2+ 12. Qxd2 Qe6 13. d5!? Qg6 14. 0-0 d6 15. Rac1 Rfe8 16. Bd3 Qh5 is the line that Fritz 13 suggested though I feel 16. Qd3 makes play more difficult for Black. It also scored the position as +1.7. When I had it play from 16. Qd3, it scored +0.62. Kf1 is suggested with the idea of artificial castling after g3, Kg2 Rfe1 and Kg1 though this seems very speculative in view of the fact that Black simply consolidates with d6, Nc6 and Be6 later. Both positions are going by tablebases and not really worth examining very far beyond the 12th or 13th move as Black simply consolidates and White's pieces are disjointed. Analysis engines are good for post-mortem study and possibly openings preperation but I don't know how many real life players would take either side in this game as it really looks from the position that no matter what move White plays, Black can only equalize.

There is one position I know of where Chessmaster 9000 makes an obvious blunder. It will literally throw away a draw by the opponent forcing perpetual just to save a few pawns.

If anyone is interested I can show the position. I'm curious to know what more up to date engines have to say.

My guess is that the Bb4 is a target. The engine must have calculated that after something like 11.Kf1 Qa5 12.c5, it will be captured by pushing the queenside pawns. Best way to know is to play a few more moves 'by hand' against the engine and see what happens.

 You are partially right, I didn't realize before that the bishop is trapped.

After the queen moves, white will play 12. a3 Ba5 13. b4 Bb6 14. c5 and white wins the bishop.