Let's look at each street. Because stacksize is unknown I assume you are ~200BB deep and you are in position, because he check-raised the turn.
AA is a 3-bet almost all of the time. You should definitely call sometimes, but the majority of the time you should 3-bet. So I like the 3-bet, but your sizing is too small. I'd like to see at least $35, but probably $40 or even $45. It depends on exactly how deep you are, but you should size up here.
This is a very good board for your range, which generally means you should bet a lot. AA is a strong hand and doesn't block your opponent's value range like QQ or JJ would, so this is a clear bet. You can slowplay other hands. I think half pot is again a little too small. You are missing out on value and giving him a great price to call with draws like KT and T9. Bet 60-75% of the pot and put him in a tough spot.
Again a good turn. Nothing changes. You still have a significant range advantage. The majority of the time you should be betting here, but $40 is a way too small and to be honest really confusing bet. You got the goods. Get some value! Try betting a size where you can shove the river and stack a queen or a jack. Even if you have to bet close to the amount of the pot. You have AA-JJ. Put him in a tough spot. You wouldn't want to bet this size with bluffs, so why do it with value?
Now having bet $40 and facing a raise this small there is really no decision. You should call and see what happens. Raising with this particular hand doesn't make too much sense.
Will he play this way as a bluff? It is hard to say. I always think it is important to play a good strategy and then (slightly) adjust it to counter what your opponent is doing.
So what kind of hands will get to the river like this? The only person that can answer this is you, as I don't know how you play. I am just roughly guessing here, but you might get here with AK, AA-JJ and a couple bluffs pre-flop that couldn't fold turn like AQ, KT, QJ, which should be 3-bet a low, but not zero percentage of the time pre-flop.
So you call with QJ, QQ, JJ and fold KT, AK. You should probably fold KK, because you block KT. So far you only have like 5-10 calling combinations depending on how often you 3-bet JJ, QJ and maybe T9. He might never bluff here, but aces just need to be in your calling range. If he has it, he has it. Playing it like this you know your strategy is sound and if you lose you'll get him the next hand.