How do you all handle a straight on the board? I have some example run outs that I'm curious about. These are just example hands, let's assume $300 effective at 1/3 live for all of them.

Hand 1:
Villain raises to 12 from utg+1
Hero flats from cutoff with JTs
Everyone else folds

Flop: 457r
Villain bets $12
Hero calls

Turn: 4578r
Both check

River: 45678
Villain checks
Hero: ?? Do I ever bother trying to push him off a chop and under what circumstances?

Hand 2
Villain limps from mp
Hero raises to $15 from co with QQ
Villain 3 bet to $40
Hero calls

Flop: 457r
Villain checks
Hero bets $60
Villain calls

Turn: 4578r
Villain checks
Hero checks

River: 45678
Villain jams
Hero: ?? Am I always supposed to bluffcatch when the preflop action makes a 9 feel unlikely, or generally am I supposed to believe a jam at llsnl and lay it down?

Is there a good "general rule of thumb" regarding a straight on the board, no flush possible, facing a small River bet vs an overbet? Thanks!

Hand 1 why would you float the flop just to check a turn that is good for your hand and favors your range? I think that's more important to address than a super rare river spot.

Did you have a backdoor flush draw?

These weren't actual hands I played, I can't remember the action now but I saw a straight on the board a while back and it just got me curious about how to play the spot if you end up there.

For the sake of hand 1 maybe let's say it was JQs instead of JT and I had a bd flush draw that missed the turn.

The point is that it depends how you got there. If your line makes sense (for repping a better straight) and also your range contains more straights than villain's range, and you think villain would never check if he actually had it, then you have a good spot to bluff him off a chop. If your line is such that he'll never believe you, or it's possible for him to have the nuts at a decent frequency, then bluffing is pointless.
Cliffs: It's not just the fact that there's a straight on the board that's important. It's whether you can credibly represent having an even better straight.

P.S. If villain jams on you when the board shows a strong made hand (meaning your hole cards have no value), the general rule of thumb is "Don't call off your stack if the best you can hope for is a chop".