Grunching

PF: standard.
Flop: your calculations are almost correct - a backdoor 1 gap straight is 1 out (2 gap is .5, and BDOESD is 1.5) making your total outs 5.5 - I personally like the flop raise because you have position, and your opponent likely does not have a made hand at this point. You can also take a free card on the turn if you miss completely or only pick up a straight draw.

Turn: Fold equity + 9 more outs + SB is showing weakness = easy bet

River: You are most likely beat by AK or AQ, and on the off chance that the SB hit the board, it's clear he isn't folding. No sense in wasting a bet - easy check.