What would we do if the Q on the flop were a King or an Ace?

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What would we do if the Q on the flop were a King or an Ace?

Blinds: $0.02/$0.05 (5 Players) CO: $5.99
BN: $33.56
SB: $6.26
BB: $3.89
UTG: $5.28 (Hero)
Preflop ($0.07) Hero is UTG with Q Q
Hero raises to $0.15, CO folds, BN raises to $0.45, 2 folds, Hero raises to $1.22, BN calls $0.77
BTN is a regular (18/15 with 3bet % = 9), he is playing 6 tables so I don't expect him to call a 4bet with less than semi premiums. I guess that he has now in his range AKo / 99 / TT / JJ. (Maybe trapped Aces).
Flop ($2.51) 5 5 Q
Hero bets $1.10, BN folds
1) Is this a bad spot to CBet?
2) What would we do if the Q on the flop were instead an Ace or a King? If we cbet and get called, is the turn a check/fold?
Final Pot UTG wins $2.38
Rake is $0.13

Thank you guys.

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