2000 Dodge Neon a/c cooling on and off. when I drive the a/c gets warm when I'm at idle it starts to cool? Any suggestions?

Nikki is the engine light on, are there any codes,have you had a tune up or any other work done lately.when you say it won't start when warm(do you mean engine will not turn over, or does engine turn and refuses to fire.
If I had to guess failing fuel pump, they tend not to build fuel pressure when ran a while(warmed up) and when left to cool build somewhat better pressure.
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Is Your Fuel Pump Going Bad
How do know if its fuel filter or fuel pump that is causing the problem
How To Diagnose Fuel Pump Failure EricTheCarGuy
http://www.clarks-garage.com/shop-manual/ts-01.htm#hot-start

Does you car have a thermostat? If the thermostat between the radiator and engine was removed instead of replaced then the warm water (which gives you heat via a coil under your dash) inside the block rushes to the radiator and gets cooled faster than the engine can reheat it. This rapid heat transfer doesn't happen while the car is idling because no air is flowing through your radiator to cool the water.

it might be what I recently found out about mine. take it to an auto parts store and run a diagnostics test and if a code p0505 comes up then it would be your Idle Air control motor circuits

i would suggest making sure you have proper level of coolant in it and maybe go with a lower temp thermostat like a 140 or 160 degree

In mornings it blows cold air even when temp gage shows warm. After 20 min of driving it gets warm not hot warm.
In afternoom it blows hot within 10 min.

try a piece of cardboard in front of the radiator to reduce the air flow and chill factor through the radiator and around the engine

You might have a bad coil pack causing the engine to stall after it gets hot, I would start with changing the coil pack, good luck, Gary

your system needs to be recharged

The starter is most likely bad and needs to be replaced,either the solenoid or motor are binding when it heats up and the parts expand,I can't pinpoint the exact cause because you haven't indicated if you hear the solenoid click wwhen you try to start it warm.I realize the starter has been replaced,but with a remanufactured unit they are not tested when they are warm and a bad unit will pass and be rebuilt.

fan motor may be drawing to many amps. see if it moves freely when not on, check for loose connections