The solvers in these spots would like do some kind of mix between both checking and betting. With the AJ it's really a mix between check and bet, this can depend on your opponents, stack sizes, a variety of things. I have found that not many people are folding to cbets in the MTTs I have been playing so that might make me more likely to check or double barrel. For the kings hand in general I am checking that turn. The solver would likely say there is some checking and some betting but I would imagine that it leans towards c/c on that turn. Mostly to make sure that you still have strong hands when it looks like you are giving up on the turn.

Its hard to answer these posts without stack sizes or preflop postions.

I would bet AJ on that board in both situations. It's hard to have a c/call range on that board because we'll be betting all of our strong hands.

With the KK hand, I c/call because it's hard to get called by worse with a second barrel there.

3 flush boards are definitely tricky and will need mix strategies.

Also depends on how connected it is. Connected 3 flush boards shold involve a lot of checking from OOP on flop and turn.

As a general rule, you want to build a bigger pot hen you have high card of the flush suit on flop or turn. You want to keep the pot small when you dont have the backdoor flush draw on flop or the redraw on turn.

But when you have the bdfd or the redraw on turn, you mix with betting more preferred than checking.

In tournaments with smaller SPRs though, check jamming turns is a viable option so you will need to think about how that incetivizes you to check more hands on turn OOP.

In general, playing dynamic boards OOP with one pot sized bet behind on rivers is very difficult.