[The Dude]

First of all, you should be 3-betting this flop. You have no guarantee that A) button will bet the turn, or B) SB will put more bets in if she does. With position you can sometimes call to raise the turn, but out of position you need to make sure your opponents don't take free cards.

The turn and river cards are both very bad, but your play is much easier if you're in the lead. Bet the turn, and if raised you can fold. Bet the river, and if raised you can fold. (This is specifically against the passive lady.)

You should bet the river, because AT, A9, etc will call but might not bet themselves, and check-folding sucks. (Check-calling is much worse than bet-folding, so that's not really an option.)