QQ sqz spot. bet OTR?

if you cbet this board 100% (which seems to be the case), you should probably use a smaller sizing.
i prefer to have a checking range and then this hand goes in it, as played turn is also definitely a check/call imo. you want to define the bet sizing and extract value from lower pairs with a club, but villain can raise decent hands here and deny your equity.
barreling the river seems kind of insane, yes we are check/folding, but we'll see a decent amount of check behinds and then we still win sometimes.

I agree with Tomaloc.
For me this flop is a check call. As played I would prefer to check call OTT and 3 barreling would be overplaying this.

It's fine to cbet with overpairs. Check/call ok, although I prefer betting too, but much smaller. Bet bigger with draws.
Turn is mostly a check/call indeed. You just don't want to get raised here with an overpair and the 3nuts draw.
River ok as played.

Agreed i also like a check call but cbetting isnt that bad you can always pot control on the turn.And you dont need to go this big you can cbet 1/3p and achive the same results.It's not a 3 street value hand though as the others have already pointed out