bet turn, he´s checking turn behind so often with some outs against your hand and he has no incentive to bet most of his hands, since they have sd value and need no protection on this board. he shouldnt bet river too thin for value either, because you most likely check your bluffcatchers on the flop. but i guess in mtt´s i would bet rather wide myself on the river, since it´s harder to figure out on what level opponents are and you´re so rarely beat. although you´re range preflop should be rather tight, so i guess i´d even check back river with AQ, since i got no bluffingrange when the turn brings a T

Mandatory bet on the turn I think. When he calls preflop 3-bet and your c-bet on this board, his range contains a TON of Ax that's probably calling again on the turn (not much draw on the flop, and you don't block Ax), but he might check back if you check, given how much range advantage you have.