yup, I just changed the 4th and final tire on my pathy yesterday. it made a world of difference in traction. I can now easily go places in 2wd that I had trouble doing in 4wd earlier with only 3 newish tires (ok, new to me).
Based on my limited experience (1 week) the pathfinder heater is severly underpowered for the cold we have been having. (I'm in chicago and we have been negative lately). My minivan actually has been better in the snow :-\
1. the build date should be on a tag in your drivers door jamb.
2. the vin will tell the dealer possibly
3. you can check the tooth shape by removing the top timing cover (I am assuming you can remove it like on the maxima)
4. 1&2 assume the motor has not been changed ever. I know of one gentleman who has a 1990 maxima (VG30E) that was swapped out for a 1994 VG30E VERY early on it's life by the dealer, so much so that all the underhood wiring was changed over to 94 spec along with the injectors. Yes it is very weird.
tstat shouldn't cause erratic idling. the idle is affected by the signal it receives from the coolant temp sensor, and the IAC. If one or both are bad you will get an erratic idle. ALSO low coolant will cause issues. In your case I believe it to be an IAC issue. the dark exhaust is a symptom(?) of running rich.
after reading this thread at least I know I'm not the only one whose pathy runs cold.
though mine constantly idles at 1k. doesn't matter what temp things are at. first start of the day, or middle of 2 hours of hours of continuous running.