First I will talk about all-in situations. Is AK or QQ better to go AI preflop?
Depends a bit on your opponents range... Against someone very tight AK, especially suited is marginally better (because AK has more equity against AA, KK-range than QQ). Almost always QQ though. There's not a huge difference though, unless your opponent's range is close to ATC, then QQ is a lot better. In any practical situation the
odds are always so good that it doesn't matter which one you have. There are no situations where you could make a decision to call an all in based on whether you have QQ or AK. The EV difference is so small that it doesn't affect your long term winrate.
One could argue that because you have one A and one K the chances of your opponent holding AA or KK are reduced, which might give you slighly more FE because your opponent's 3/4-bet is a
bluff a bit more often.
However, I believe the slighly increased FE is negated by the fact that sometimes people call with 88 just because they think you have AK - and when you have QQ instead you crush them instead of flipping. So they might fold a bit more often when you have AK - but when they call light you have more equity with QQ.
Post-flop play: An overpair is a more disguised hand than TPTK. So I prefer QQ. Also AK can't make a set, which is a very strong hand. QQ can.