[locutus2002]

I'd lead the flop. Against so many players it is unlikely that hero would check the flop if he held a jack. Once hero checks, his hand becomes very vulnerable OOP. Villain's 1/2 pot bet into a 4-way pot looks fishy, I might consider c/r to T1800 if I had a read on him as a joker.

The turn was very lucky for you. Hero would probably have to give up on many turns. I'd push the turn.
My guess is that hero is behind against villain's range slightly.
~30 ways good jack
~30 ways good overs
Villain is ahead slightly more pairs than hero is ahead.

In spite of this, I'm calling 100% of the time on the river after the turn, given the overlay in the pot.

Villain's 15% equity when hero is ahead is substantial because of the implied odds.

Pushing the turn is also consistent with a variety of holdings for hero.

I don't like villain's line. I prefer checking behind (preferable because WA/WB) or push the turn. His 1K bet is supposed to look like a milking bet for a jack which is now much less likely and the 1K bet commits him to the pot. He probably makes a slightly EV- call here against a c/push by hero.

After the river, I call with incredulity and curse the poker gods.