Running Rich?
 

I have a 92 s13, stock KA. I noticed the car doing this a few months ago.

When I sit at idle for a few seconds.. maybe 15 seconds, I can rev up the car and it throws a LOT of gray smoke out.. it's a lighter gray. I read online that it may be running rich, but I have no other symptoms than that. The car runs great, and has no noises or other problems at all.

The only thing I can thing of.. The O2 sensor wire hit the header and the insulation got burned off a bit, but the actual wire inside is fine. Could this be causing it or could it be something else?

Blow-by. Check to make sure your PCV valve is not stuck open... Other than that, it could be tired rings or valves/stems/seals giving out.

I'll check the PCV. One more thing, I forgot about this.. I just did a 5 speed swap. The car was smoking like this before the swap though.

Anyway, I can't believe I forgot this but what I was mentioning was if I rev up the car in neutral, it goes down to idle at about 1300 rpm for about 2 seconds then goes back down to normal. I thought this was maybe just how an auto ECU worked because I've never owned an auto car, but maybe a vacuum leak could be causing the problems?

A vacuum leak will cause idle issues etc for sure... but not any crazy amount of smoking. My 240 was an auto/manual swap with the exact same thing you are describing before I did the enigne swap... Bottom line is that your motor is probably high-mileage and tired. Great time either for a rebuild, 'new to you' used engine, or a swap. Take it in for a compression check... The numbers will tell all you need to know.

Alright it actually has about 150,000 miles which I thought was pretty low but maybe this will give me a reason to swap it. Haha I appreciate the help man.

Well before you go and replace the engine you should take care of some of the things that would make it run rich. Coolant temp sensor, clean the maf, replace o2, and run some seafoam in a tank of gas. My car was runnig rich too and i did thise things and it's smooth as hell now and running right. Like $40 total for all that too.
What do your plugs look like?
Gray smoke is usually fuel, blue is oil.

Btw i think they all do the idle thing you described. Or maybe try cleaning the iacv.

Last time I changed my plugs they were completely black. They were in for 4 months.

Thats running rich then. Really rich too. Do the things i said inmy previous post. Also you might want to pull your fuel rail off the manifold but still leave everything connected and watch each injector spray as someone cranks it and make sure they are actually squirting fuel and not just sraying consistsntly.

I bet oil is getting either past the rings or thru the vavle stem seals.

Op is the car getting low on oil?
Gray smoke is almost always fuel. Valve guids being bad would mean a nice puff of blueish smoke at startup. Rings would show it more at high rpm. Check for these things.