Well, 10.Bxc6+ is an illegal move, but I see what you mean. After you take the rook his Bxc6 is annoying because your bishop on b4 will also be hanging.
So after he loses the right to castle you probably play something like 10...Be7 and you're better because he can't defend his rook.
The other player resigned with 13,5/15 minutes left on the clock.
Afterwards I actually didn't feel like being winning even after taking his rook. I wonder if I should have castled instead of 9. ...Nc2+?
If 9. ...Nc2+ is the best move, how do I solve the situation after 11. Bxc6+?
Also, if I made obvious mistakes before 9., please let me know.
Thanks!