Preflop is fine, 3 betting is also good.  I prob opt to call majority of the time.

I dont liek raising the ass end of the straight draw on flops but his bet is very small and your call is going to allow BB to call wider.  However we also have limited bluff out lines.  Getting a decent Q to fold is going to be hard, and we are not raising all that much on the flop outside of 44/QJ and draws(in some world we raise AQ here if we flatted pre).  So I would opt to flat a decent part of my range here, 9t inclusive.

The turn deciison is interesting, BBs range is easily the weakest as he is prety capped and had great odds both times.  Main villains range is very hard to estimate.  He shouldnt really ever be super strong though, so our range can be the strongest.  He also probably doesnt have a good draw or I would expect another bet.  I think my read on villain is JT/AJ/KJ and QT/Q9 type hands, and al lhis hands we expect to fold.

Do we have to bet this turn?  Yea probably.  We have a lot of equity, we cant really rep(I guess we can have some Jx) any river we miss(if they both check again).  We also have the very bottom of our range here, well not anymore but on the flop this is the bottom of our range, we have an OE with only 1 nut draw.  Must of our other hands are going to be actual pairs or strong flushdraws and KT(which has two nut draws as well as an over card).  A lot of villains will expect us to raise strong draws on the flop so our range is weighted more towards pair and weak draws, so betting 100% of our hands balances nicely(especially because our weak draw nuted the turn without acutally hitting).

Sizing is important here though.  Villain is either c/calling or c/fing, there isnt really a middle ground for him since we can expect all strong hands and all flush draws to continue here on turn.  So I say we bet like 8.5k.  BB is super capped so and in a different but similar result position in that he cant really afford to call turn without his pair hands(because value willl be extremely ahrd to come by on river). 8.5k is also a good amount ot bet with our Qx type hands for value, and even with a strong KJ or AJ.

Oncevillain calls the turn, the river is pretty interesting in that we expect his range to be very capped, but also the 9 probably hits his range more than our range(he can have J9/Q9 more than us, but we can have KT a lot more).  What value hands do we bet on river?  KQo(I assume we 3 bet KQs pre a decent amount), AQo occasionally, KT, and probably QT/Q9(although Q9 pre is marginal). We can also throw in a hand like 8Tss or even J9ss.  So theoretically we can have a lot of value here(although I dont think some of these hands bet 11k on turn).  But we still need to get villain to fold.  I assume we can get all Js to fold and get looked up by most Qs.  I think a bet on river is the way to go, but has to be a value-y bet as again that 9 isnt amazing for our range.  I would proably bet like 26k.

I made the mistake of not expounding furhter when I said either c/c or c/f(although I do think he is pretty much never c/r'ing).  What I eman tby this, mis that there are not many border line hands that may call or may fold.  He isnt really waiting to see what you bet, he is either giving up or plannig to pretty much c/c any reasonably under pot bet.  When hand ranges are like this we should really be betting towards the samller size, even with our value, for a lot of reasons.  Betting bigger will put us in awkward spots and make river value harder, and in the case he does c/r.  We also have to think about our entire range, we dont have too many super strong value hands here and villain knows that spo our bet should as a whole be favoring our range.

In regards to your first question I think the hands he is most liekloy to c/c down is going to be QTs or Q9s.  Occasionally see AJ/KJ if he is likely to hero.