After the Flop, checking to the 3-better is standard as is the QQ's bet. UTG has 10 outs (3A, 3K, & 4Q) plus a back door flush and it is not sufficient to call on odds alone. Before calling, UTG needs to know what his plan is on the turn.
As played, when the Ace comes on the turn I don't like the check. I would prefer to bet 1/2 pot to make button
As played, I agree with the 1/2 pot bet on the river and the crying call by the button.
Hero is holding AKo. Villain is holding QQ.
I think it isn't a cut and dry thing.
I think your play is right at least some portion of the time...and some portion of the time you should be betting the turn.
What rate? I'd say bet it 75% of the time and check it 25% of the time.
But if you do decide to bet...what will you do if the villain comes over the top?
In general you played the hand well. Taking into account that the villain 3-bet of 5x (2.86x pot) we can play calling or 4-bet. The important thing to say is that the times that we are OOP the frequency that we bet 4-bet is very variable compared to a wider range position like BTN. And if you call with AKo you don't always have to, as you have to take advantage of your range advantage.
The flop is standard just like the turn. Actually on the turn you could have a donk range. Because if you've ever played slow with a: AK combo or a: TT combo, you'll want to get some value back. If you've ever played loose preflop, now you want to print full speed with a combo: A5. And if you ever want to render bluffs in this texture, you could do it with 3 combos of spades like: K9s-KJs.
On the other hand, your villain could call with an off AQ combo, which delayed the action, he thinking of his blocker's advantage.
Also the villain could raise sometime with a nut combination (QKs) or with a combo: AJs. But the main idea of donk betting is to target the closed range of the villain which is 9 combos: QQ + KK. Maybe some of them are part of his fold equity range, but sometime we can give it a try, when we let V represent the dominant range in preflop.
Another option as played, is to make a bigger bet on the river, as we've delayed a lot of action on the previous fairways and now villain retains a good part of his dominated range.