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- Do I need to purchase and install an additional gasket (Part# 35692)?
It should come with the gasket. The therm. I got for my KJ came with the gasket. The gasket might not be on the thermostat but in the box. Check the box for it, if its not in there check the thermostat when you take it out.
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- Where does that gasket go? In between the thermostat housing and the thermostat or in between the thermostat and the engine?
It goes around the thermostat forming a seal between the therm housing and the block
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- Will installing the thermostat backwards cause me to have not heat on A/C and only blow cold air?
It won't allow any coolant through causing an overheating
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- Will installing the thermostat without that additional gasket cause it to leak coolant?
Yes. Get the gasket and also get some gasket maker. I used blue RTV on the block for the thermostat I just changed on my YJ. Some say its too soft and I should have used red RTV or super gray, but its been fine so far and I've put 200 miles on it already. So with the RTV and gasket it shouldn't leak.
But your friend was incorrect, the spring portion goes inside the engine. Just make sure its seated correctly and don't get the gasket wet or it may ruin it. And CLEAN THE BLOCK AND THERMOSTAT HOUSING!!!! Get ALL of the old gasket material off, I suggest a single sided razor blade like this

Once its all off, then go ahead and reinstall the thermostat and gasket and you should be good to go.
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'02 Jeep KJ 4x4- Gone
'92 YJ- lift, swampers, bumpers....yeah, its pretty sweet.
*DISCLAIMER*: I can not be held liable for anything I say on the forums or in real life.
I also can not be held liable for any jeep breakage or life problems caused due to my advice being taken. 