I like the river shove, I don't think he is gonna have 2 pairs here much, if ever (56 hitting on the turn being the only possibility, specifically 56 of diamonds), sets are unlikely with the bet sizing, he would either bet super small to induce a raise looking like he is really weak and just wanna see a turn or he would donk out a lot bigger with a set, turn again he would bet bigger if he had a set, that's not a turn you try to milk on, he is betting like he has a hand, but that he isn't ready to commit with it himself. I would shove the turn here as the hand has played out given his sizing. He could be on a draw himself which you would want to protect against and it can also look like you're on the draw by shoving, I just can't see how I am beat on the turn as the hand played out.
I see T9s, JTs, QTs, KTs, ATs, possibly any Ax FD, maybe some K high FD or even Q high seeing as there are no A or K on the flop. The only had that beats us is a weirdly played set and I don't think that will happen too much at all, MAYBE 34 of Diamonds to see a cheapish turn is possible, but again the combo's are so few. Also if he was slow playing QQ+ here he would for sure bet bigger on the flop if he is gonna donk out. To me his bets simply look like a TP that wants a cheap showdown in case he is up against overpairs or have a draw drying to get there as cheap as possible and then bomb it if he hits.
What does he check/trap river with OOP that has JJ beat with less than potsize? Wouldn't you rather shove for value against a stubborn TPTK type hand than give it a chance to check back? The river shouldn't change anything from the turn as hero never has a 4 in this spot? He is checking to catch a bluff or give up on his draw as he has no room to do anything else with less than potsize bet left, I can't see how it's ever a check back with JJ here as with stack sizes being what they are you can't afford to check a really good hand here and have a TPTK type hand check back cause of a 4 card straight that neither player is ever likely to have