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Well ...

The tire's being different diameter [if they are or were for any length of time] cause essentially damage the clutches and or viscous coupling in the trans. There is a thread on here with pictures of replacing the rear section which has those parts in it, you may be able to source one at a junkyard.

I would try getting all the tires the same first and see if the problem goes away since any variation is interpreted by the car as a wheel slipping and will engage the lockup.

Can't say if the smoke is continued trans leakage or remnants of the initial overfill. There are a couple vent tubes on top of the trans where it would puke out fluid and there is a chance that the pump housing is cracked and forcing fluid out that vent.

Shouldn't be any connection between the CEL and the turn signal. Check the turn signal bulb housing for corrosion [especially the rear] to see if there is a ground/hot loop being created.

 

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It's not so much the manufacturer but the diameter which varies between manufacturers.

I've had over half an inch difference on different brands, brand new, with the same number stamped on the tire.

Mixing and matching tires is not nearly as important on non fulltime AWD vehicles as far a handling. The only issue on a 2wd would be ABS and possible differential if it were LSD and the tires were different side to side.

 

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Yeah, That's the thread I was talking about.

I'd agree that you need to figure out exactly what's going on with the trans before deciding to throw money at it, might be better off with a salvaged replacement if it's too far gone.

Like gross gary suggested, there may be a blown seal between the trans and diff. That would explain trans fluid loss with little or no external leakage.

Check the diff fluid level, if it's high, drain some in to a white pan and see if it's a reddish color. If so, might want to think full replacement.

 

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No air filter ? nice.

No telling what the previous owner did to that car as far as fluids, cheap "fixes", wiring etc.

The guy might have tried to pull the dash apart to roll back the ODO and screwed up some wires or connection somewhere. Completely unfounded accusation there but, I have a huge problem with people selling things without full disclosure and, doesn't sound like he disclosed much if anything. Telling you there was no filter in it would have been a good start.

Go down under the dash [next to the fuse box]and see if anything is piggybacked off the turn signal relay. Turn them on, you'll be able to hear it clicking.

If not, if there is anything like aftermarket lights or stereo equipment, check that wiring to see where it's getting power.

 

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Tube that ends over the exhaust ? is it a tube that goes side to side [boot to boot on the rack] ? some racks have a equalizer tube to give the air in the bot somewhere to go.

Sounds like the previous owner didn't have a clue what they were doing.

As far as fluid transfer from the diff to the trans or vice versa, that would depends on the size of the leak. You can watch both and see if one drops and the other rises or if the gear oil starts to look pink.

You can try some Lucas heavy duty oil stabilizer in the diff or Limited slip additive to see if that quiets down the whine enough to think about fixing the torque bind issue. I'm thinking no but ....

If it keeps whining, I'd think about getting a replacement trans/diff.