AA and KK are more in your range then opponents so I would bet halfish pot. With a paired board its doubtful a FD is raising here oop and even a 2 is dead to KK, so if we get raised its an easy fold imo.

I agree with all of that. But if I get check called on the flop what should my line be on the turn?

Another way of asking the q is how many bets do i want to be making with my AA, AK, QQ, JJ in this scenario?

If I get check raised on the flop with that range I am usually behind and will likely fold.

If I get check called and then checked to on the turn, my instinct would be to bet again. I could balance range by checking back some of my AA, AK on that turn and betting some. If I bet 100% I open myself to a lot of check raises on that turn with FDs, trips, SDs, etc.

But then those situations are always kind of gross because if I check back on turn and he pots into me on river I need to be calling it off probably 30% of the time on a blank river, right?

Checking back is fine with QJss in a 3-way pot. You can't go for 3 streets of value when hitting the flush somewhere, and you need very little protection. Also, getting raised sucks because you wouldn't want to fold, but calling is meh at best.

Agree with midori. You're not accomplishing anything with a flop bet. Never getting anything better to fold, very little protection, very little value, and you risk getting ck/r off your decent equity by a naked 2 or a bluff. This is a pretty perfect hand to check in order to protect your checking range on this board.