Without doing any math, I would guess that if you're playing JTs, KTs, KJ, and heart draws like this and jamming river then you should be jamming AT as well or else it seems like you are likely overbluffing the river.

Flop+turn looks good IMO. If you check on the river it can be very tricky, as he has a lot of value combo's and not that many missed draws/floats. I like to be aggresive in this spot and just jamming the river, although this would be the bottom of value range. 

I hate this spot. In general, I x/c turn and x/f river. Seems very weak, but probably the line were I lost the less money in the long run.

His range for calling turn is mostly Qx, Ax, and hearts.    Your 3b range likely has 88-JJ, and you are probably barelling the A turn.  I would bet a bit outside my value betting range and fold to an unlikely shove.  Maybe 3/4 pot.  I think you can get called by Q's and worse A's which are checking behind.  You have to protect your barreling range on the turn with a river bet unless you are check folding AK/KQ here which I think is bad.   It's a good  board for your 3 betting range I would be surprised if he turns hearts into a bluff here.  If he has you beat he will let you know.   I guess it's exploitable in that our range is pretty capped but given the 3b flop turn line I doubt a reg will shove over our third barrel here.

I dont think there are a lot of Ax on his range as you block Ah that most Axo who might float would need, and turn is Ad - the bnfd that he may float you with. I can see him floating flop with 2 backdoors (suited broadways of d), and just calling down with some monsters, so his range has some strong hands - I dont really think you can check/call in this spot. I think the really tricky part is what is the absolute worst calling hand in his range, and from that point just count combos and try to find if you can valubet this or not. IMO this spot can be  just an annoying check/fold.

Probably check-call river, but it might be a check-fold because you block quite a few of his bluffs which would be the nut flush hearts which didn't get there.

I dont think the missed nfd has to bluff river, it still has some shodown value. Most bluffs are missed hearts with no SV and Qx turned into a bluff, and without more info nor balance considerations I think this is a CF river.

You need to have a turn 'x/not fold' range, and AT is a pretty good hand to put in it in spots like this.  But I think I'd bet your hand here because the Ah blocker decreases his Ax floats on the flop (making his range weaker) and the Q on the flop blocks AQ which is a big part of his non BFD A hi flop floating range.  So, on this runout I think I'd be betting turn for value.  On the river you should check and call, or check/fold if villain is very nitty (and jamming if he's a station).