This is 5-handed so the BB's standards for a 3 bet are considerably less. I think he has A-something (not AK) and whiffed the flop.

I agree that raising the turn is not hot given the flop action since you will win more when ahead and lose less when behind by calling. However I would bet the flop and the turn if checked to. Your hand is definitely worth a showdown after pairing on the turn.