OPtion 3 is probably the most OPtimal:

Pot Odds: ($56)/($336)=16.7% Odds to call OTT

Outs to hit Full-House+ OTR

3(A)+3(4)+3(9)+1(Q)=10 Outs for us OTR

(10 Outs)/(46 Remaining cards)=21.7% times we hit OTR

Clearly Calling Turn is the most OPtimal Play while we can 

1) Fold on all Unimproved rivers

2) Call With 100% Frequency when we hit our Full-house+

There is as small problem. I was trying to think about his range he can have like 13 flush combos 3 set combo (i assume he would've raise set on flop). So its a tight value shipping range for him

But we still beat 3/16 combos (18.5%) and we have to be right like 17% plus he can have couple % of bluffs and thinner value like A9,AQ,A4... who knows

OP, ur giving VIllain too wide of a hand range for MR u OTT.

A4, if probably too thin of a raise on 3-flush Board.

no. I said even w/o this thin combos we have 3 sets of 99. And unknown bluff %. It can be quite valuavle mistake to fold to the price of 88 to win 513. So maybe shipping turn is better

Now i convinced that pushing turn OOP is the best option. IP call-call is better

OP, Ur basically piling Assumptions after assumptions of WHat VIllains Turn Raising range is, which i think is a mistake without solid reads to back-up your statements. 

*) I'll analyze the worst Scenario for your QQ VS Flush hand range equity Wise:

Text results appended to pokerstove.txt 308  games     0.000 secs    61,600  games/secBoard: Ad Qd 4s 9dDead:   equity win tie      pots won pots tied

Hand 0: 77.273%   77.27%  00.00%           238        0.00   { KdJd, KdTd, Kd9d, Kd8d, JdTd, Jd9d, Jd8d, Jd7d, Td8d }

Hand 1: 22.727%    22.73% 00.00%            70        0.00   { QcQs }

1) 

All-in Pot:$512.16      Remaining Effective stacks:$145.58

EV(call): 78.3%=I Lose     21.7%=I win

($512.16)(.217)-($56)(.783)=$67.3 +EV on our Call if 

We're folding all Blanks, while calling when he hit

EV(Jam)=($512.16)(.227)-$145.58= -29.3 EV

SO unless VIllain is Jamming with Wider than FD, then clearly Jamming turn is -EV. 

because on turn he can have hands weaker then ours

and pot is big enough

i totally agree with TheLove_Below,i thins we have to analyze it in the worst case scenario,bcz is the most common,if u are saying he is a fish,and he is playing all his draw passively,i think u must remove the set combos from his range,99is pretty rare and the other set could be just 44(this kind of player usually raise the flop when it's drawy and he is holding a vulnerable hand)

(as usual sorry for my english,i hope u can understand me with a loooot of imagination ;)