I would never ever check on the flop. We're the preflop agressor so this is a dreamspot for us cbet as we're expected to bet a lot of the time, and villain can easily assign us a wide range (but still narrowed down given our position of course).

Turn is fine, but river bet is way too big. Given how the hands was played it screams that you want to extract a lot of the value you missed earlier on. So to sum up, bet flop, bet less river.

Confused over how the action went down in this hand - did btn call pre for an all in?  

You can't check this flop. I know there are spots where you should check back with the nuts, but this isn't one of them. You should only check back the nuts if there are hardly any hands the opponent can call with. The opponent would call the flop with any reasonable Jack, a king high flush draw, a wheel ace - there are hand that your opponent would call a cbet with, but would not bet out if checked to on the flop. Sure there are some hands that will bet when checked to, a lower flush for instance, and there are some hands which will fold to a bet, but you wouldn't have got action out of them anyway, so what's the loss? All in all, you're missing out on some serious value on the flop. 

As it turns out the turn bet was fine, but I agree that the river bet is way too big, it just shouts value and your opponent will fold a lot of hands they would have called a smaller bet with. If you want to get value on the river, its best to bet out before hand and bloat the pot to the point where your opponent is more likely to be "committed", with their weak equity.

I'd cbet it, because J is in the range of the vilain, and he can also have a heart card.

In the turn, i think it's okay check behind, cause you give him the opportunity of bluffing

Bet/check/bet seems most ideal.

I LOVE bet/check/bet here. 

Yeah, bet/check/bet looks good because we wont get 3 streets of value from bare J almost ever and because this runout also often improves his range so hes more likely to pay off river.(even though we very rarely show up with a bluff here).

I also think that on this board people wont be stabbing too much vs missed cbet(not without a heart and will also check mediums hearts back imo) so we dont gain almost anything by checking the flop(Kh or Qh could be stabbig but its also calling a bet so...)

Imo in this spot the most profitable line it's

bet/check/bet

Prolly sizing something like 1/3 flop and 1/3 river getting value from some Jx (most should be improved in 2P turn/river)

Have to bet the flop. On that turn, I'd bet/bet here. You want all the hands that have Kh or Qh and maybe a little bit of something else to call another street before the river bricks off. And part of his range improved to top pair, two pair or a gutshot.

If the river is a heart you get value from Kh/Qh and the gutshot/2pairs/TPGK aren't calling anyway. The bet on that river seems fine for size, slightly player dependent - but they either have enough to call or they haven't. Unless villain is a chronic unbeliever in which case either an overbet or small bet is best depending on his exact variation of stationitis.