I like a fold, hard to see how you can have the best hand here on the turn.

Don't agree on pre-flop. Your hand isn't so smooth that you can pot/commit on a whole lot of flops (and expect to dominate your opponent's stack-off range). Instead I think it's best to just flat, and try to play a small pot most of the time.

As played I'm not sure. Since the board is actually pretty dry, but also is unlikely to have hit your 3-bet range, I wouldn't mind a c/c. But I guess that also leaves you open to getting barreled on turns and rivers. I think a lot comes down to opponent tendencies. Myb against someone passive you can check, and against someone (overly) aggressive it's best to just go ahead and bet yourself (and probably again on the turn). I'd rather avoid these tricky spots though and 3-bet hands that are a lot smoother from OOP.

Villain is aggressive 37/21/6 regular with CO RFI 35%

you probably meant BTN open

gave villain $FI40!4b3
flop countinuation range: 34+,5+:68+ (can't think of much else)
which is 26% of his total range calling pre.

So, betting w\ no equity here should be auto profitable line (untill he floats wider\bluffraises)? or I'm missing something in this sub-range?

Feels like we have small equity advantage on this flop

well, questionable, I gave same sub rage for hero's ($3bo8)
and we only match 34+,5+:68+ 18.5% of the time

I disagree about your pre-flop. Against regs, I like to xR a lot on BB or donkbet because they just fold, they are so tight. Because I flat a lot I think that I need to have some strong hands in my flat range and this hand is perfect for this. With a T or even a 9 I would accept more 3bet, but the 8... two gaps? I prefer flat.

The flop I don't think that we need a lot protection. The hands that hit this flop, have something else, like a pair and he probably will raise us, like 875X.

So I think his float range is stronger, like A2 or 2p+ so it's very easy check fold OTT. Even if it's not, we are playing a fucking guessing game OOP.

A(3) 2(3) d(10) 6(3) 7(3) 8 (3) = 25

25 cards put you in trouble, in other words, 57% of the rivers. So yeah, it's a fold.