Preflop: I would flat the open vs unknown as standard but since you are deep and OOP 3-betting seems a fine option also. I figure villain will call IP wider than if you were both 100bb deep. Although we have to expect him to start calling with a ton of speculative hands and worse Ax before we get towards 50% equity vs MP alone. I think we would have to expect to get him to fold often postflop if we take this line. I'm not sure that's what we want to be doing here with AQ.
When BB cold calls OOP I find this suspicious. BB could just flat with some hands he thinks have good implieds and is not concerned with MP clicking it back ( with something like: 22-JJ and AJo+, KQo+) or he could be trapping hoping to induce vs MP (with something like: QQ+).
I would expect MP's calling range to be something like:
Spoiler
Equity Win Tie
MP3 52.93% 49.11% 3.82% { 22+, ATs+, KTs+, QTs+, JTs, T9s, 98s, 87s, AQo+ }
SB 47.07% 43.25% 3.82% { AhQs }
Flop: In a multi-way pot OOP my decision here would be between bet/folding with the intention to continue on good Turns, either by 2 barrelling or x/c'ing depending on the card. HU I think we could even x/c because the board is dry, villain might stab as a bluff and we may have the best hand some of the time, but multiway I think if we check we have to check/fold as I wouldn't expect either villain to bluff too much.
As played....
when we c-bet I think we definitely get 22,44-99 to fold sometimes villain might fold a Tx. Particularly BB b/c he is OOP vs MP and may get squeezed. I don't think we get any Jx to fold OTF or OTT unless they're particularly nitty, just because I expect both villains calling range OTF to be pretty strong just b/c: 1) you have 3b pre = strength 2) c-bet into two opponents OOP. Additionaly, most of the Jx and Tx they're taking to the turn will have some gutshot / redraw ( eg, QTs, KTs, ATs, AJ, QJ).
I think the decision OTF to c-bet and to barrel (as a bluff) OTT would be very opponent dependant. If villains are folding lots to c-bets OTF then their Turn range is likely strong so I would lean towards giving up on the turn unless you improve. However, villain(s) may be the type to fold TP to two barrels on a blank run out but you would have to have a fairly decent sample on the their fold to TCB IMO, but as a default I would lean towards checking and folding in this instance.
If villains have a low Fold to FCB% their Turn range is likely to weaker and include some more 2nd and thrid pair type hands. Therefore I think we can be more inclined to barrel as a bluff even on a blank run out like 2♣: for example. The 6♠: isn't obviously as good b/c it puts a FD out in case villains called OTF with some backdoors but would still be okay IMO.
To sum up. In this instance I don't like the turn barrel b/c villain(s) ranges will be so strong OTT and I don't think we get much to fold here or OTR.
River: Weird card really b/c now you have TP, but like you say there are very few hands (if any) that call a value bet OTR. Since you bet small you might get a really Loose and passive opponent to call with QJ/KJ/QQ. But I think most opponents won't pay anything off with worse. I think x/folding is best here unless you know villain is the type to call 2 streets w/ a draw and bluff missed rivers.
Sorry for the essay. Can't help myself.
