I would prefer to 3bet or fold this pre because if u flat the UTG this pot is most likely going multi way. Your hand plays pretty bad multi way. 

Also why aren't you betting flop once PFR checks? It's not often you can have a strongish hand on this type of board, you have a tonne of equity now. Good to bet it imo.

Since UTG doesn't bet the flop and you don't eitehr, its harder to narrow down his range, but you think he would cbet the flop with any sort of heart draw, especially the NFD. Once he calls the turn I'm more likely to put him on clubs and some sort of combo of straight draw+club draw or pair+club draw. 

I think you can bet river for value but also don't mind a check too much if u think he  might put u in a tough spot. It's kind of thin though. You basically have the nut low hearts and are probably only getting called by 78. He doesn't have many if any lower hearts in his UTG raising range if he is a decent player. You seem to have some sort of read on villain that makes you think he might c/r bluff this river? What makes you think this?

whether you bet call or bet fold depends entirely on your read on him. Do you have any notes on him?

fold pre, bet flop. probably check river. I think lots of his range contains flushdraw+pair ott(either club or heart) so that I don't see much value of 7high flush.. especially when you fear c/r shove as value/bluff. 

Dont think we get called by worse OTR, maybe some loose a4xx but since we hold two 4's that's unlikely. I would check behind.

fold pre for sure, check back river as well

Often in these spots i find it that villain either has a set or getting stubber with some hhxx hand. I dont belive sets are calling the river and flushes obv calls. But i think ppl more then often donks if they hit there flushes in these spots rather then go the c/c line since we should be checking back the maturity of our range on the river.