The first thing that stands out here is that a guy sitting on 41 blinds opens from the btn and you flat call out of position with QQ. This is uh to put it one way non standard so I assume you must've had a specific reason to play your hand this way against this particular player? Perhaps you just never 3bet from the blinds versus steals so doing it here would give away (or possible even over rep) your hand. You said this was a live poker hand so I'd be 3betting it regardless though.
Checking the flop was obv with the intention of check/raising I assume. You may want to consider leading here on this Q low flop despite flatting pre as villain will figure to be either way ahead or way behind and thus refrain from firing a cbet, and there will be more hands in his value range here that call or raise a flop donk than he'll bet himself if checked to. If he has demonstrated a tendency cbet a lot (i.e. always or when he misses by a mile) then check/raising as opposed to donking flop is fine.
The worst result you could have envisioned occurs when the player checked behind and the
peels, for reasons which should already be clear. I agree with checking here a second time, as painful as it is that you were dealt QQ with 40 blinds effective (or less if you don't cover), flopped a set (on an A high board at that), and it now looks like you can't even get stacks in because there's been literally no betting yet post.
The river is a bit interesting once the turn checks through, as is your bet sizing. It's not a spot I'd ever find myself in because I wouldn't have played the hand like this, but I don't mind a river bet. I suppose I could state the obvious and say villain's own game is key here and whether or not he's prone to bluffing
in this spot, but after checking twice in position this hand already as the preflop opener, the odds of him betting the river with a hand you can beat don't seem great. Calling a jam is, again, villain dependent, and you could make a good case for folding your set (as well as using more typical sizing otr). It's extremely unlikely for him to shove worse here for value, but on the other hand he could certainly be spazzing/bluffing quite a bit given the board and the unusual action thus far and that alone may merit a call.
Overall this hand was somewhat unusual, and it all stems from the unusual play preflop by flatting QQ this shallow.