My 1994 Toyota Land Cruiser ALSO idles rough in cold damp weather.

The Idle air control valve ( IAC ) Gets dirty over time and needs cleaned from years of use. Remove and clean carbon from it, Also clean throttle body with throttle body cleaner. At most auto parts stores Advanced, Auto zone, ETC.

new spark wires. old wires hate, cold damp weather.
spark plugs, cap,rotor wires, try the 60k mile tuneup. per the opr.guide.
magic happens....
the map is ok.
and to test a map you need the graph and you need to compensate for altitude and local weather (barometer readings) to correct what you see.
is the check engine light glowing running?
most maps read like this, 1bar maps.
http://www.fixkick.com/sensors/my-map.html#Jeep

i bet your's is ok. the map.
your car is a toyota corolla in snake clothing.
the toy FSM covers this is find details.

put a trans funnel in dipstick tube and ad fluid

Best not be an azzclown and get your T Belt changed and it may be less than 1500 dollars but good luck with that,ya get what you pay for!

sounds like a air leak on the inlet manifold ,allowing air in which is causing a weak mixture which is causing the overheating?' possibly

Had exact same problem after driving 120 miles in Florida heat. After 30 minutes started but ran rough for a day and now seem fine. Thought at first fuel filter or dirty gas. Going to replace filter anyway. Is this a common problem?

replace the IAC valve to resolve this issue.

If it is happening more frequently in cold or damp weather it could be a spark issue. Wet or damp weather can effect spark plug wires if there is even minor damage to them (usually occurs durring rain). The other things that come to mind are the fuel pump and lastly the fuel line check ball that is in the fuel tank. If it is leaking then all the fuel in the line will drain back into the tank over night (pressure lost in the fuel line) and this will cause repeated, extended cranking when you first start the car in the morning or after the car has sat for several hours.

I think you are low on refrigerant in the system, you may have a leak, you will need to hook up pressure guages to determine the exact cause.