Light 3bet flops TP against Nit OOP

Button is 20/15 with a 70% steal from button and 80% fold to 3bet over 350 hands. BB is an aggressive 7/8 nit. I was thinking BB range is 99-QQ, AK, maybe AQ.

Preflop: I raise because button steals and folds to 3bets alot.
Flop: I bet to get value from 99-JJ
Turn: I plan on check-folding but couldnt resist a call after his small bet.

Did I play this hand fine? Should I be Cbetting? Should I just fold turn?

No-Limit Hold'em, $0.10 BB (9 handed) - Hold'em Manager Converter Tool from FlopTurnRiver.com

MP2 ($20.87)
MP3 ($9.25)
CO ($10.01)
Button ($15.16)
Hero (SB) ($10.53)
BB ($10.63)
UTG ($2.64)
UTG+1 ($4.43)
MP1 ($10)

Preflop: Hero is SB with Q, 9
6 folds, Button bets $0.30, Hero raises $1.05, BB calls $1, 1 fold

Flop: ($2.50) 7, Q, 5 (2 players)
Hero bets $1.30, BB calls $1.30

Turn: ($5.10) 5 (2 players)
Hero checks, BB bets $2, Hero calls $2

River: ($9.10) 10 (2 players)
Hero checks, BB bets $6.23 (All-In), Hero folds

Total pot: $9.10

whats his turn betting range?
99-JJ AQ/KQ and flushes.

I don't think he would fire the river with JJ or 99.

I think you played it fine.
Although I am not sure about the turn.

Flop seems fine, I maybe c-bet a bit larger, but I don't really know.

Turn seems like a check/call rather than a check/fold. His range as you see it is something like {AhKh, AQ, 99-QQ}. If you bet then you probably fold out 99-JJ a lot, whereas when you check he may well bet here with those (and you have more than enough equity to call).

When he shoves the river I think we can safely take 99 and JJ out of his range, which were the only two hands you were ahead of so the fold here seems fine to me.