Would you fold the straight on the river?

I would fold here. I wouldn't spend too much time in thinking can he bluff here or not. You have invested 0.13$ so I would just let it go.

do you like the conti-bet here?

It usually doesn't hit my MP raising range except overpairsI think

I do not c-bet there. It does hit villains range much more than yours. Also really nice board to float. You will not generate much folds here by cbetting and your gut shot is indeed not much when you hit as you found out.

Normally ( if we forget about this board for now ), I wouldn't cbet these small underpairs especially OOP. I also would disagree with floating on most of the boards because we usually just open these small pairs to setmine and when we don't hit we can just give up.
But in this case you have a gutshot also. Of course our straight would be at the lower side and might not be good. But if we want to play this hand I think I would rather bet once and give up than check call.