QQ vs preflop action

I think 4bet is ok.
OTF I don't see much reason to cbet on this board, except it will be a info bet (and that is not a good reason to bet).
When he calls your 4bet he likely has Ax hands and TT+, so pretty much only hand in his range that you beat is JJ?

I think it would be best to x flop and make a delayed cbet OTT if he checks behind. As played give up OTT.

I concur.

If you held a 4-bet bluff hand that completely missed (if you have those at all in your range), then bet and rep the A. Betting here would be just "bet and hope he folds". Even a call by a worse hand (better hands aren't folding) is bad news, since you WILL then usually get bluffed out.

I'd play check-fold. More courageous players would be able to float once OOP here, but I am such a wimp. I see his range peppered with Ax, and if not, I see him double barreling. I simply cry "mama".

The chance in winning this is hoping to see a cheap showdown with mostly checks by the two of you - maybe including one check-call (if cheap).

Other option is flat 3 bet
Check - call flop.
On the turn he should be wary about what you are holding, and might check behind. In that case stab 40% river.

4bet sizing is not good, they have full stacks, we're inviting action with decent enough hands from villains and be out of position on the flop. I would shut it down on the flop and try to get as cheaply as possible to the river, and folding to any decent bet (1/3+) most likely.

I would 4 bet bigger. OTF I like more c/f. The fact we are OOP sucks...