Bet turn - you don't want KhX AhX or like JJ/TT/99 with a heart checking behind.

I can't really put villian on a hand here. He could have 2 hearts, AK, a Q. I can't believe he has a straight on the river (would have to call twice and bet with 8T or KT!, and if it was hearts, he'd already have a flush.).

I think you can check the river, but not intending to fold. You showed weakness on the turn, so you are inviting a bluff.

If you bet turn and he flat calls... I think then a river check is in order, or a blocking bet if you think he will only raise with the nuts.... tough river either way.