66 BvB faces minraise on 567r

His minraise can be some draws, overpairs, weaker or better sets, any 2 pair, made straight...
I would try to stack off there. I think 3bet folding wouldn't be good. He can stack off with weaker hands, like overpairs with a draw (88,99) or with 55, and even against a straight you have equity.

These minraises can be anything. The nutstraight or any kind of hand that has some perceived good equity for the villain. In my experience it's better to just raise and getting it in on the flop under 100bb. If we just call, generally on the turn the villain makes a small bet and we have the same information. It can be the nuts or any other good enough hand from the villains perspective.

If you 3bet don't fold to a shove. I would just gii , he's got lot's of 2 pair and pair + strait draws aswell as 55 and some over pairs

BvB in these situations, I think it's best to 3-bet the flop and jam most turns. I think playing a 2 street game at these stakes/on this texture is the best play, generally. As you move up in stakes, you could do other stuff on the flop (such as c/r-ing or using a checking only strategy OTF).

Should we also consider that there are no flush draws on the flop, so wouldn't villain be slowplaying a made straight most of the time? Does this weigh his holdings to being drawing hands and 2 pair type hands? I'm inclined to 3bet here and call a shove.

The argument for the 3bet / fold is that I don't think he will stack off anything worse than 55. Maybe sometimes he has 2 pairs but I feel like I'm ahead of his reraise range and way behind his 4bet range.

This being said, I see the value in 3 bet / jam. I'd prefer to have a more agressive opponent though.

I'd just call the flop, and check call next streets. I only stack off if the board pairs.

You should not fold a set in this situation!
If you decide to 3b, then you should stack off.
I prefer just calling and seeing the turn.