Sometimes I play the same way. More often I 3-bet the flop instead of waiting to turn. Would have been interesting to know more about the player. Some players play almost any hands from BB knowing it´s going to be played heads up. Ax is then a possible holding.

Betting the river is a bit dangerous since u risk two bets to win one (If u´re check-raised u must call, even though u´re almost certain to be behind). So u need to be a lot more than 50% sure u´re ahead (2/3 to be precise). Think both check and bet can be correct. Depends mostly on your opponent