In a ten minute blitz game, my opponent played an unorthodox opening. On move 9, i gave up my castling rights, but my king seemed well defended and my attack eventually broke through. My question is: Was forfeiting my right to castle a mistake, or perfectly acceptable for the position?

It seems like the mistake was your opponents...getting that deep inside, that early, without support to back the move seemed strange.  If the middle game had been stronger for your opponent I might side with the idea that you shouldn't have given up castling rights, but with the Knight being played that aggressively by your opponent without support, I think releasing your castling prospects was a safe move.

I think it was a mistake. Allowing Ng2+ and giving up castling is no big deal, your king is perfectly safe. However, allowing your opponent to trade that poor knight for your e3 bishop under favorable conditions can't be that great, and eases his position somewhat. White is still clearly better though, 3 pieces and a rook developed to 2 pieces, with kingside attacking chances. I would say that playing 0-0?! is dubious for black, as defense is difficult and he is castling into your attack. He should instead castle queenside later. Then he proceeds to close the center. When one side is attacking on a flank, or when one sides king is in the center, the other side should open the center, not close it, and he might have had some counter chances.

If allowing my opponent to trade of my knight was a mistake, how do you suggest I should have avoided the trade?

I'm not sure, I don't believe I would actually retreat my knight, but I wouldn't allow him to trade his for my bishop, just my knight (especially if I knew he would play 0-0). That bishop is pointing towards the kingside and can help with an attack, and although it is technically your bad bishop it is a strong piece that will become active. I think even Kf1 might be playable, preparing Be3. Bd2 seems a little passive, but is also an option.

It's said that after Ng2+ Kd2 0-0 your bishop is trapped, as his knight is forced to retreat to h5 and may even become trapped if a trade was avoided at that point.

*Bishop, there is no other good development square. Also, if he traded, my once f-pawn does a tremendous  job preserving my center and I would rather have the knight in a closed position.

I'm probably wrong, but it looks to me as if Black still had a defense as late as 22...Rf5 instead of 22...Qb6? And after 18...Qa5 instead of 18...c4? I think Black is a bit better

You're not wrong, both better moves.