Hand 1:
If I didn't think he had an ace, I would call the minraise and give him the rope to hang himself on the turn and river. He may have a weaker king like king-jack and I would be more suspicious of paint on the turn and river.
Hand two:
I'm check-calling the turn or possibly betting 1/2 pot to keep the pot small. You have kicker problems here and this is often a situation I have been in with AT against hands like AJ or AQ and my players are too passive to raise. River is bittersweet because you're not folding trips.