AKs 3bet MP vs UTG vs 1/2 pot bet OTR, flush board

Terminology is wrong. It would be a cold call, not a limp.
3bet is ok, I would mix in some calls too.

I think the cbet OTF could be a bit too large. Continuing OTT could be a decent idea. You could get one more call from weaker AX hands and from some pocket pairs that have one spade.
As played I think the fold OTR is ok. Villain looks tight and I expect him to mostly call your 3bet with suited aces and quite good pocket pairs. So he can definitely have AQ,88,99,QQ here. And flushes. I'm not sure if he will be value betting ATs or AJs.

Preflop I think you can get away with 3b only, no need to call. Flop is too big. The best would be 1/3, 1/2 should be fine as well. Turn is weird, if u bet smaller otf, i would just cbet 40% (or smthing around that). However as played pot is already big, so not sure if we can bet. I agree with lassi about the river.

Pfff probably wise to fold OTR but hard to navigate as played, your flop sizing too big OTF puts you in deep.

Indeed. Cbet smaller, you have position. I think you can bet/fold turn in most situations. He's probably looking for a check-raise OTT knowing you can have all the Ax and would bet again if he has the flush. Once you check-behind and he bets again, he probably has you beat, giving you a good price to call. Some sets could bet/fold that river, in case you're slowplaying a flush OTT. JT seems a very long shot, only see one combo here.
Villain seems a decent player. I'm not sure about his bluffing range in this spot, but I think he doesn't have that many.
I'm curious, so getting 3.1-1 it's close. Once it is NL10, the nitland, fold might by better.
Without the A♠: I think raising OTR and re-open the betting round is not a good idea as I don't like to turn our hand into a bluff when we have showdown value.