I'm not really in love with any of your options on this flop so I guess the check is reasonable. I'm not a fan of the turn bet though. Any card between Q and 6 will likely be better for your opponent than for you so starting to bet here isn't great. If you think villain will bluff you a ton on the turn then I would rather take a call line than to try to bet him off his bluffs. That line doesn't sound great without any info though. I think AJ in this spot might just fall into the x/f bin and if villain checks back letting you realise some equity then great!

I generally call the AJs here but have no problem with the 3bet. I don't think betting the flop is a disaster, just a bit thin but after checking the flop I don't like to bet turn here, there isn't much we can to fold to one bet.
If we do go for the bet turn though, I think we should also bet river on, otherwise it's a one-and-done scenario which just gives our opponent the opportunity to fire anything at us on the river when we check it and we can't do much about it. Basically, betting turn here then checking river is just giving our money away imo.

Once you skipped the flop bet (which I think is correct btw, though betting probably shows surprisingly similar expectation in the long run) I don't see much merit in bluffing the turn. You might even have some showdown value with this hand.

Calling or 3 betting is fine pre. Villain looks a little on the passive side so could be the type to defend a lot of combos of weaker Ax making a 3 bet good.

I like the flop check with what might be the best hand but can't really extract value.

I used to give up a lot in this spot no the turn but had surprisingly good results making a delayed c bet and firing most rivers. Against a lot of players you will have plenty of fold equity versus second pair type hands which assume correctly that you never bet worse for value.