I don't like villain's line at all. Since the flop has been addressed, I'm going to provide my thoughts on the turn. The optimal turn line is bet/fold, in my opinion. When villain is ahead, you'll have ~10 outs (villain is likely only ahead of AK, and possibly AJ), and therefore it's paramount villain charges draws, and avoids giving a free card.

I also think there's a strong argument for check/folding the flop because of the reverse implied odds. It's likely villain is behind, and when ahead you have a vast number of outs to improve to a superior hand. Conversely, villain's drawing close to dead, and it's unlikely you'll pay off with a busted draw/overs.

Come to think of it, I'm leaning toward simply folding the flop.