Hand 1:
I'd rather re-pop Villain on the flop and fold to the push (if it comes). This is the perfect flop to make villain define his hand, the only decent draw is QJ, if he pushes you can lay it down and feel confident he had you beat. If he calls, he's probably holding a draw or a 1-pairish hand. Either way, if you then check the turn, more than likely villain will check behind and you may get a cheap showdown. If villain still bets on this turn after increased action you can lay it down confidently.

Hand 2: Fine.