I think a check raise is actually an interesting idea on the flop if you want to go that route. I think check calling and trying to hit the straight isn't the best option. If you hit the straight on the river and bet out, it turns your hand rather face up. Unlikely villain is super strong when there are so many two pair and straight combos he likely checks behind the river if you check the nuts. Since the button will raise wider and almost always CBet on this type of flop, I think a check raise is an interesting idea to fold out a lot of his range. That way if he hit the straight, our hand is very disguised and we can pick up a heart on the turn making it a very profitable spot to double barrel

X/raise or call OTF, fold OTT, 9T comes in + potential bd flushdraws, so continuing might be very tricky, so I lean towards folding on this texture. Any 9 cannot be treated as such a live out because of its missing nut potential on 4 straght boards often shut down the action.

I like a x/r but ofcourse we don't need to go overboard and x/r too much here.
As played just x/f the turn. It's not a great card because a 9 isn't that great anymore for your hand.

I like xc otf, I'd be check raising a lot more polarized
xf this turn