98 Dopdge Caravan won't start when temperature around freezing . When outside temperature rises one or two degrees it will start on first try

I assume from the question that the engine is cranking over but will not start. Hence there is no issue with the starter motor on a very cold start.

At 40 degrees F you are getting closer to freezing. When this happens again I would check the operation of the fuel pump.
Take the fuel cap off the fuel tank and have someone switch the ignition on and off a few times. (just the ignition - don't start the engine up). Turn it on for 10 seconds or so and then off. If you listen at the fuel filler opening with the cap off you will be able to hear if the fuel pump starts and stops. If it is working it should start and then stop a few seconds later each time you switch on the ignition because the computer shuts it down in a few seconds if the engine is not started.

If the fuel pump is operating I would be looking for some electrical gremlins. You would need to check that the fuel injectors are getting power and that the spark plugs are also.

Ideally you would need to a quality scan tool connected to the car when you have this very cold no start condition whilst trying to start the car to see what faults may be present because it should pick up some malfunction if the start problem goes away when the temp rises a few degrees over 40F.

The air from the heater core will be as hot as the coolant temperature of the engine. 80 Degrees is the opening temperature of the thermostat and that will be where it remains until the engine starts to work or the ambient temperature rises.
Thermostats come in a couple of different opening temperatures so check around for one that opens around 85 degrees if you want hotter air in the cab
Another point is to close the recirculating vent door to the cabin as the heater core is trying to heat fresh freezing air if it is open

dont know the full location but its on the front end around one of those ares are that you have had replaced, its only smale and has 2 wires connecting it, it clips into an area were there is good air flow.

I would suspect the thermostat first.

Possible bad cold start valve or the sensor that tells the valve to activate when cold. Start by replacing the cold start valve first.

Sounds like the fuel pump has an issue with a safety on the pump itself.

You probably have a failed fuel pressure regulator.

at 100 degrees c the aux fan will come on. Make sure that fan is working properly and also that your fan clutch is working properly.If this is happening with the a/c on there is an updated temp switch for aux fan operation on the receiver drier. It should be red in color. The temp is going to rise under different loads but that is a bit extreme in my opinion. Make sure you have a 50 50 mix of water and coolant and that the rad. cap seal is o.k. assuming the rest of your cooling system is in proper working order.

sounds like a faulty outside temp sensor. This is located behind the lower front grill on the rad support. This sensor is likely causing your both issues as the voltage drop or resistance is not in the range the ecu expects. You can check http://forums.fourtitude.com/zeromain they will help to clairfy swapping the sesnor if you feel like getting your hands dirty.

Try changing the fuel filter, water trapped in it can freeze. Its a 92 and probably time for a fuel pump. They don't like water either.(Meythonol Fuel) Always change fuel filter when adding antifreeze to fuel