1) Just fold PF, or at least reraise. Calling only "makes sense" if you are doing it as a complete bluff and are planning to push him off the hand but as played you seemed to have called for value.

Against some players this may be semi-ok, but you need a lot more than a dozen hands on them. Also, how does the first hand really change this one? Regardless of the results I dont like this at all because you dont seem to have a read that would make this ok.

2) Ditch this in the BB, as played bet the flop harder. Bet the turn and hope to get it AI, otherwise just push the river if he calls. If you really put him on a Q, then I dont think this is horrible as he probably calls, but I just dont like how the hand played out to get there.