How can I tell which transmission I have?

No place I bought it from put a 6” lift and 35x13.5x15 tires on it but didn’t change ratios and dropped transfer case which causes it to pop out of 2nd and reverse.
 
Welcome to the forum!

Even after regearing the diffs, depending on what you plan to do, the 35" tires are probably a bit much for your TJ unless you have the optional Dana 44 rear axle or some aftermarket axle upgrade. If you are not planning on going offroad, your probably OK but then the 6" lift would definitely be a tad overkill.

What are your immediate plans for your new baby; street machine or rock cruncher?
 
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No place I bought it from put a 6” lift and 35x13.5x15 tires on it but didn’t change ratios and dropped transfer case which causes it to pop out of 2nd and reverse.

If you want to keep the lift you'll need an SYE
 
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Little of both as it’s gonna be a daily driver but planning on upgrading to better gear ratio thinking 4.11 and in the process of finding a SYE and CV rear axle.
 
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Idonno, I'd be really hesitant to daily drive a Wrangler with a six inch lift...unless it were a nice one. Does this lift have adjustable control arms or track bars? Are there pucks (rubber spacers) above or below the springs that you could remove to lower it a bit?

With 35" tires, you'll probably also want to go even deeper in your gearing, especially if you really want to go off-road with it. But, again, running 35's on a Dana 35 and doing any serious off-road stuff is probably asking for a snapped axle. Short of getting a beefier axle (Dana 44 or Ford 8.8), you could upgrade your axle shafts to aftermarket chromalloy to improve your odds...or do a super 35 upgrade.
 
I’m thinking about doing a super 35 mostly cause an axle swap it out of budget for a little bit. And I did a transfer case drop and figured after putting a front bumper and winch and rear bumber and some skid plates the weight would lower it some