JJ top set vs river straight in check-raise pot

I would squeeze PF, even if you got a call you're in position, it's a good spot to do that.
On the turn, i would check too. On the river, since villain checks again, i think he doesn't have a straight. I would bet here for value

Probably also would have 3betted/squeezed pre-flop and my ideal outcome would be for MP to fold and CU to call my 3bet , because CU's range will be wider/weaker and MP's would be stronger. If MP called and CU folded then you should proceed carefully.

I think you played correctly on flop + turn ( that donkbet of villain usually indicates a hand like pair + draw ). In this case he had draw + overcards which falls under the same category.

The river I also agree with mafioso03 that due to the fact villain checks river too, that it's almost certain he doesn't have the straight. The best option would be to bet very small like 1/4th of the pot so you can still extract value from 2 pair type of hands villain could have.

Re-raising the villain in order to go all-in as soon as possible, we get flat called which makes me believe he is on a draw. The turn completes a straight so decided to shut down. Can I continue betting here or are we never getting called with less?

BB leads into 4-players almost a pot sized bet then calls big raise and you think he has a draw? I think its more likely he flopped a crappy 2-pair.put a note on him that he plays bad, him having AK here is horrible play

BB leads almost pot size in to 3 players. It's a 4 way pot. Than a semi bluff would be weird for sure. It's more likey a 2pair+ hand and raising isn't to bad for value and protection. If he has a straight already it's to bad.

Off course I would 3bet JJ all day pre-flop or the MP villain is extreme tight.

Cheers,
SDK1987