Should I be betting or checking turn here? Why?

I bet. I'd rather bet turn myself than chk and call a river lead. And I'd feel compelled to call a river lead on most rivers if I checked.

Bet, Tx raise flop some percentage of the time you get value from draws and 7x,also when you bet he will chk river and we get sd.

This

bet like 60 to get some value from weaker 7x and draws, would barrel bigger with air and shippity ship rivers.

standard double barrel IMO. it's not often a T and you can't afford giving a free card

What if I had bottom pair in this spot. Say it was A2. Should I barrel as well?

yes

With BP A2 I am barreling and also shoving most rivers to get overcards, draws, 7x and worse 2x to fold.

With A7, I am barreling turn and likely checking back river for sdv.

Bet again on turn.
Guy would probably raise with Jack, so you almost always have the best hand. You don't want to give him a free card with 2 hearts on the board and he might have a straightdraw with overcards on your 7. You get value if he has a weaker 7 and he might call with a hand like K2.

what do you guys do with a very weak 7? like 73

doesnt make sense.when you chk river with A2 you win worse 2x,over cards,draws,and 7x should fold turn.

???