ive been through hell and back with my radiator.
Basically the radiator and coolant is a closed loop system that pressurizes. The stock radiators are half plastic and are pretty cheap. Mine literally exploded on the trail. (pic to follow)
The signal from the sensor on the thermostat housing is sent to the computer and it gives it reading on your dash. now from what ive seen in my 02 is that the dash gauge is practically worthless. Multiple times my readings went from 210 to redline in a matter of 2 seconds. I replaced the sensor checked the wiring multiple times. with no improvements.
Since your system is acting up its best to flush and renew the system. it sounds like your overheating badly if the motor at idle is surging and your alt output is dropping. even though your gauge doest say it is.
either you have a big air bubble in the system or your tstat is not opening and closing properly.
If i was you i would go through the system dump new coolant in it along with a new cap and t stat.
or this can happen to you
my system overpressurized from over heating due to a bad cap and my sensor reading not working like it should.
I now have a all metal radiator, new cap and thermostat. autometer water temp gauge. a flexalite black magic e fan. make sure you bleed the system really good. good luck.