I have a 2002 Prizm..Drove the car all day & parked it - no problem all day. Now, it starts, runs at high idle few a minutes like normal, then drops to regular idle, but all of a sudden, the idle is rough. Drove it down the block & check engine light came on. Turned around, came home & parked it. Where to start ?

All the above is normal , except 2000rpm is a little high at idle . can be adjusted down to 1500rpm. the computer adjusts the engine speed . in drive the transmission loads up the engine idle drops to 1000rpm . in park engine unloaded rpm increase all normal.

Don't know.
You would need to have both components tested. It could be a dead cell in the battery. Or a bad connection. If the alternator were at fault it should discharge when the engine is running and slowly go dead.

idle air control valve is bad replace it. at start up choke control runs till warm up while driving throttle opening controls engine at idle ,the idle air control valve.

First question is what constitutes "overheating".
Since the car has been parked for years, the thermostat could be sticking closed.
The cooling system could have air pockets and need to be bled.
The water pump may be corroded inside and not pumping enough volume.
If the gauge goes above half way, the fans should start up. But half way should be more or less normal, not hot yet.
I would look at flow problems first, beginning with removing the thermostat and driving it without one for a few days. Also check and see if there is a process to bleed the cooling system of air.

sounds like you need to replace the temp sensor

check and or change the plug wires also check for a crack coil pack which is highly likely on that model and also check the ignition module as well

you need to replace the throttle position sensor on the intake.

check your IDLER

I had the same problem with mine. I installed a fuel pressure gauge, and drove around until the problem started. The truck should be running between 48 to 56 lbs. of fuel pressure. Mine ran at idle at 42lbs. , not even high enough. Drove it for about 20 minutes, and it started to stumble. Fuel pressure dropped to 30lbs, 20 lbs - then all the way down to 2lbs. I shut it off, and it would not start back up. 30 seconds later, I turned the key and it came up to 10lbs. press. 10 more seconds 20lbs. After almost a minute, it came up to 42 lbs. started and I drove away. I replaced the fuel pump assemb. on 7/3/2008. Drove it around,and the problem is gone. Mine is a flex fuel, and the pump retails for $668.00. I did find a AC Delco pump for $221.00. I hope this helps.

vacuum leak..look to area behind air cleaner to intake .. snorkel area first..spray carb cleaner there after you start it to discover vacuum leak.... this is a guess .. but a good one... check first for hoses off/ leaking intake air sources and let me/us know