7 handed, hero opens LJ BTN calls BB calls.
Flop Qh2d2h, BB checks hero bets BTN folds BB raises hero 3's BB calls
Turn 3h, BB checks hero checks
River 9d, BB checks

1) Is there any hand that is close to correct to play this way?
2) What hands should I be VB'ing the river with given the action up till this point?

1) completely dependent on the opponent and dynamic...
2) same as #1, and I assume you mean what are the weakest hands one should VB here, which again, depends on the opponent...

Need more info

Agreed, is villain super FPS and you have some sick read on him, where he jams flops w/ monster draws and then check raises when he gets there? If so, unless you have a flush as well (and I've no idea why you'd check the turn then), a boat, or quad 2's, I don't see how there is a value bet to be had.

Also, this hand smells like a lower pocket pair from you (55-88 or so). I might be tempted to fire here if he's looking me up w/ a naked ace, 3x, or 44 every time. If he gets away from hands like that, or will only call w/ a pair +, I'd probably just check behind at game speed. Haven't stoved this, though.

Did you have a/k or a/j with one heart?

Is the villain going to call you with no pair on the river?

I'm assuming you have JJ-44 and checked the turn afraid of the C/R (so you have a heart, right?)

As played, I think I bet my whole range here for value if I'm right and you have a pair. But it obviously depends on the villain.

If he does, the flop 3-bet is pretty bad.

QTs/Q9s?

1) Q4-Q8

2) TT+

Yeah I can't really come up with a credible hand he would play this way and that was the only thing I could think of.

Q7s

I can't do it any more, and I'm not satisfied.