16NL FR - KK in 3b pot on paired board, I get lost on the turn... HELP!

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16NL FR - KK in 3b pot on paired board, I get lost on the turn... HELP!

UTG: $16.48
UTG1: $16.75
LJ: $5.36
HJ: $10.32
CO: $16.24
BN: $16
SB: $17.23 (Hero)
BB: $16
Villain is running 12/10 over 510 hands. His 3b stat is 4% and his showndown range is QQ+,AK.
Preflop ($0.24) (8 Players)
Hero was dealt K K
UTG raises to $0.32, UTG1 folds, LJ folds, HJ folds, CO folds, BN folds, Hero raises to $1.12, BB calls $0.96, UTG folds
This is the first time he's cold called a 3b. I'm not sure if we can rule out the top of this player's range, or whether he would just flat QQ+,AK here? I doubt he has AQ. So my thoughts are he has 22+ (he feels he's getting good odds, especially if the original raiser calls) and AK.
Flop ($2.64) Q T 6 (2 Players)
Hero bets $1.63, BB calls $1.63
I make a standard value bet on the flop (how's my sizing?). On this flop I'm not sure there's much that I get value from... I guess maybe JJ, but that's pretty much it.
Turn ($5.90) Q T 6 Q (2 Players)
Hero checks, BB bets $3.16, Hero calls $3.16
Given there aren't many Qx hands in his range is check/calling okay here?
River ($12.22) Q T 6 Q 9 (2 Players)
Hero checks, BB bets $10.09, and is all in, Hero folds
On the river I think he either has a boat or complete air, with the boats making up the vast majority of his range. I don't think he jams AA here. I guess he could have AsKs as well. So now I'm not sure whether folding is good on the river or not.
Final Pot
BB wins $11.59

I think the check/call on the turn is fine as he could be trying to value bet his lower pairs, but when he jams the river I feel like I'm beat a lot of the time. His river AF is 0.4 and AFq is 22% so I think this is very unlikely a bluff and is a shove for value.

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