1) I would play as 100% call against big lead sizings (over half pot), but in general the solver approaches small leads with a more merged range and my experience analyzing these spots requires us to raise quite aggressively against them. I don't know exactly how my opponent is approaching this spot: it's very possible that he's being very polarized using a small bet strategy, therefore the best option would be to just call, as equity denial becomes less of a concern. I just relied on my intuition based on studying turn leads in BB vs BTN scenarios and solver response on various textures (even those in which the OOP player clearly has an advantage in terms of nutted combos).

2) Flop I don't mind his play, I think having 5xss is even better to put this 3bet in, as you also have a blocker to the best set I can possibly have given the preflop action, but nut FD + gut with a deepstack is a fairly reasonable combination to consider 3betting with (especially cause I'm not only raising sets or monster draws there). On the turn he's representing a very polarized range (sets, monster draws, set blockers), so I expect him to continue with the aggression, especially because he can improve to the nuts both on an offsuited 2 and on a spade (he also has the Ace of spades, so less chance to see the board pairing in case a flush comes), so I would bet around 2/3, 3/4 of the pot. River check call is completely opponent dependent.

QQ+ can comfortably shove. An argument could be made for AJ, but I would just play it opponent dependent and shove against some/check against others.