Big draw / what the best play?

MP1

H 24
VP 25
PFR 13

BB

H 130
VP 32
PFR 6
DB 36 (14)
AF 2.5
WTSD 29

Notes

min bet with TP no kicker / middle pair
limps OOP pre with AT
slow plays a big hand with a wet board
will auto jam a draw
will call a pot bet with middle pair

Merge No-Limit Hold'em, $0.04 BB (7 handed) - Merge Converter Tool from FlopTurnRiver.com

Hero (UTG) ($4.40)
MP1 ($2.80)
MP2 ($2.52)
CO ($4.49)
Button ($4.69)
SB ($4.39)
BB ($3.14)

Preflop: Hero is UTG with 10♦, J♦

I don't want to discuss if we can or can't raise this hand preflop. This is a LAG style play. I feel it would be a waste of time to argue about it. Just know I'm not playing this hand to hit a J or T and i'm not playing fit or fold post flop.

Hero bets $0.12,
MP1 calls $0.12,
4 folds,
BB calls $0.08

Flop: ($0.38) K♦, 7♦, J♥ (3 players)

BB bets $0.04,
Hero raises $0.25,
MP1 raises $0.70,
1 fold,
Hero calls $0.45

We are getting 3:1 on our call so this is an auto call. I have 9 to 11 outs. If we go with 9 outs then I only need to make a little over 1/4 the pot to break even.

Turn: ($1.82) 8♥ (2 players)

Hero checks,
MP1 bets $1.40,
Hero ???

At the time I checked because I didn't want to get check/raised off my hand. If a J is good as an out then I have the odds to call but I would be pot committed and a J maybe no good so I think a fold maybe best. I have 0 fold equity so I don't see the point in shoveing the rest in. Now that I think about it if I would have bet 0.98 then I would have an auto call. The pot is $1.82. He has $1.98 left. A 1/2 pot bet would be $0.91 so we make it a little more so he has $1. Our odds would be 1.82+0.98+0.98+1.00/1.00=4.78:1. If I put out a weak bet like $0.98 I would say he will just shove the rest.

I think his range is:

77
JJ
KJs
AQd


Maybe
AK

I think he would 3 bet AA and KK but there is a small chance he may have them.

Thoughts

The crucial point in this hand is when he 3-bets you on the flop. At this point you really have to lay out your plan for all the rest of the hand, not just calculating odds for each street in a vacuum.

I pretty much agree with the range you put him on, except I don't think many players at this level would 3-bet a FD here like that (and of those who would, many would simply shove), so at this moment you should realize that you are behind the vast majority of times.
For the rest of this analysis, I will discount the possibility of him having a better FD than yours, as I find that very unlikely for the reasons stated above.
Also at NL4 I would expect to have close to zero fold equity in this spot, so you can't expect to be able to push him off his hand with a raise on any street.

So to conclude on the above, your problem on the flop boils down to whether or not you think there is enough equity in taking your FD to SD, hoping for the flush to complete, which is pretty much the only realistic scenario for you winning this hand.

Therefore, when you call the flop, you should be planning to take the hand to showdown, otherwise you should have folded.

Now for the rest of this rant, I am not going to do the exact math here, as I can tell from your post that you can do that better than me , instead I will apply the approximate reasoning that I think it is realistic to apply when at the table, in 20 seconds, in the heat of the battle:

In ballpark numbers, after you call the flop, there are like $2 in the pot, and he has about $2 left in his stack for you to win if your FD completes. This gives you roughly 1:2 incluing implied odds, which is about equal to the odds you get from your 9 outs to the flush.
Let's also say that the discounted outs due to the fact that he may have a FD also, are outweighed by your backdoor SD.

Conclusion: This is a coinflip. You really have to have a good read to be able to weigh this above or below 50/50. And from 24 hands, you don't.
You may be able to do the exact math and add some detailed reads to make this a 49/51 in either direction. Me personally, I would fold to his 3-bet on the flop on a bad day, or shove (and take any fold equity I can get, however small it may be) on a good day.

But I really consider this a coinflip situation.

It may not be a huge factor, but none of your outs are clean. So if you caount them as if they were, you're over-valuing the strength of your draw.

Is this a shove or fold situation on the flop? I lean towards folding, since we don't have any FE and can pretty much assume we are behind.

We need Deco to come in here and run some numbers for us, see where we stand against MP1's range.

would like to see this players stats , if your reads are correct about this player than hes not 3 betting on the flop but will auto jam his draw. So our outs appear clean. I put him on KJ, set, AK and alot of crap like 9,10. k7s AJ, KQ .
I realy want some evidence that this player is solid otherwise im taking this all the way to showdown at these stakes , fold flop at higher stakes and against good players. Seriosly the amount of fish at these stakes just make this hand too good to throw away , jam the pot on the flop as calling only pot comits us anyway. most players at these stakes do not consider pot odds, fold equity and pot commitment .

stick it in on the flop, barring that, stick it in on the turn.

I'd pop it up to about $0.50 OTF and happily get it in after that.

*EDIT: I know you said you didn't want to argue about preflop, but it's a pretty clear cut fold imo.

if you don't wamt to stack this board then fold preflop. You say you are not playing fit or fold postflop but then you flop a shitton of equity and don't really know how to play it. The problem you have is you can't really draw OOP as your implied odds suck big time -- he is going to play his hand a lot better than you on the river -- so calling flop/calling turn is just spewing.