Couple HU hands and a couple HU PF decisions.

Hand 1 is played fine. There is no reason to check the turn behind, because you want to steal pots like this when you have J9o or something equally random. If you check the flop behind it's not going to trick him into calling down with nothing, and limits your future opportunities.

Hand 2 I'm not as sure what to make of it. Villain is playing pretty strangely, but has given every indication that he is showdown bound. It's hard to have that inclination with a hand worse than K7s, and bluffing a missed heart draw isn't going to happen more than 1 in 5 times. The jack isn't a great card for you either, since it's an overcard that villain could have been (incorrectly) drawing to and now hit. I check and fold.