You did not play the pot correctly, you managed to lose the maximum amount possible.
Your pre-flop raise was OK.
With queens I would much rather have two aces hit the board then one, but that only means my hand is a little less likely to be beat at this point because there is one less ace in someone's hand. It does not mean anything other then that, you still do not know in the early position that you were in, if someone has a hand.
Which brings us to the follow up bet you made. The best way to figure out if the player has the ace, is if he calls a big follow up bet or not. You already know that since he is in the hand and has called a pre flop raise, that in his range of hands a ace is somewhat likely. The only hands that this player is not going to fold here are pocket pairs and hands that contain an ace. The player may fold some pocket pairs here. The flop is very dry, so there are no draws out there, the only sure thing you know before you make the follow up bet is that any of your opponents with an ace are going to make a call.
Your follow up bet is usually a bad bet, and the times it is not are because nobody has flopped anything. Even if your hand was a winner on the flop, you would never loose as much checking this hand down on this very dry board as you did on this very dry board by betting half the pot. I would have checked this flop. I would have tended to fold if the player bet. The only time I would have called this players bet is when I had a good read on the player, especially against an unknown player.
What were you thinking when the third ace hit the board? I suspect you though he could not possibly have the last ace in the deck and you put him on a pocket pair. Bad thinking because all the signs are that he had an ace, and just because the odds of him having an ace are way down, that has nothing to do with the fact that he has an ace. When your bet got called on the turn, you should have been finished with the hand. Your bet was horrible. Your call was even worse. You married your hand and ignored all the signs that you were in a terrible marriage.
The way to play pocket queens when an ace hits the board, is to slow way down. There is very little upside to betting, you can only get called most of the time by a better hand. By checking if the opponent decides to be sneaky you may get free cards that win you the pot. (In this case you would not have unless you back door a straight.) With the exception of a pre flop raise, there is nothing you did right in this hand. Your Grade is D-.