200 - 4b spot w AA on monotone Board: Strategy Question

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200 - 4b spot w AA on monotone Board: Strategy Question

CO: 227.05 (Hero)
BN: 220.70
SB: 211.90
BB: 203
Preflop (3) (4 Players)
Hero was dealt A A J 5
Hero raises to 6, BN folds, SB folds, BB raises to 17, Hero raises to 52, BB calls 39
Flop (117) Q T 7 (2 Players)
BB checks
Hero ?

Common spot where most players pot it on the flop without thinking twice. I am just curious if this is really the GTO play here? Probably biggest factor is how wide villain calls when we shove. 

Q1: Anyway has anyone played around with those kind of spots and maybe solved them coming up with ranges for bet/calling and bet/folding (100bb)? 

Q2: For example: What do you think about betting ~30 and folding when shoved on (with this particular hand)? 

Q3: How wide should we call a shove here in villain's shoes?

Q4: Does our strategy (as aggressor) change when the flop is instead 2c 3c 7c? Or when we have Ac/Kc?

You don't necessary have to answer to all questions to post in this thread though :)

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