where did I go wrong?

argh, white plays the Ruy Lopez, then after a series of moves I (black) end up a whole piece down, a messed up pawn formation, and without any clear idea of what to do next

The simple answer is that 10. ... Qf6 loses a piece tactically because it exposes your queen to a removal of the guard attack. The positional weaknesses in the opening were annoying, but the game was very much up in the air at that point.

What should I have played instead of 10...Qf6?

White somehow transposed to 2 knight defence where you had an extra tempo (with a6) to play:

EDIT - I was working something else for 15 minutes, so when I finished and uploaded the diagram, some of the questions were already answered.