These are the types of flops I just know I'm gonna get check-raised on.

The problem is we have no draws, if we had ANY face card with the aces I'd be much happier. More than anything his hand is going to be big pair with broadway cards, even if it wasn't suited, having the Ace of diamonds would be a big help as well.

If we're not behind already we are in bad shape. I can't think of any hand that isn't ahead or smashes that flop completely.

Checking back the flop and If he bets the turn, I'm more likely folding than anything else

Yeah I think this is standard backchecking too. We wont push EQ vs his range and will be way behind his c/r range.

What if we are oop?

In position, I would probably just bet-give up depending how often he is calling c-bets or raising c-bets

I would start with a check OOP too.