There are 2 coolant temp sensors.
1 wire SENDER only used for dashboard temp gauge
2 wire SENSOR only used by computer
The 2 wire sensor would be the one that could cause the computer to "think" engine is cold all the time and set high idle and rich mix.
That would cause a CEL after 5 to 10 minutes through, the computer times engine warm up, if temp sensor doesn't reach a preset minimum in a preset time it will turn on the CEL.
The IAC Valve is operated by a Step Motor, a step motor responds to voltage pulses, like a digitally controlled motor.
Computer can send IAC valve a pulse to open valve 1/4", another pulse to open 7/8", ect....I think there are over 100 pulse commands the computer can use to set IAC valves air flow.
When you unplugged the IAC Valve it is closing, but not all the way, idle should drop down to about 500 or engine should stall, either is fine.
But at 800 it means either IAC Valve is not closed all the way or you have a vacuum leak.
The unchanging idle as engine warms up could mean the step motors internal pulse recognition has failed, so motor just turns valve all the way open when any pulse is received, or the computers voltage pulsing circuit has failed and is sending "open valve all the way" pulse all the time.
I would remove the IAC valve and make sure it is closing all the way when unplugged, if not then you will need to replace it, you can try cleaning it but...........
Then plug it back in and turn on the key, it should open all the way, then unplug it again and listen for any odd sounds when it closes.
These do not fail as often as you think, people just tend to replace them when there is an idle issue, lol.
So getting one from a wrecking yard is not a bad gamble.
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