I need a scenario to help me understand. Power is applied to the unit, its remains closed for a specified time then opens until power is cut again?

This timer begins counting when the power goes off. It will not close until the set time has run out. If the power has been removed for longer than the set time, it will close within about 2 seconds when power comes on. This also prevents some wear tear on thermostat contacts (if they are mechanical contacts) because the final closing (making of the circuit) occurs in the delay timer, not the thermostat. Once the power is interrupted for any reason, the ICM203 opens until the set time has run out. Hoping this helps.

Is this a 12v unit suitable for automotive applications?

No, this unit is for alternating current (AC) use only. Automotive applications are always direct current (DC).

Will this hold up to protect a full-size refrigerator compressor?

According to the ICM 203 manufactures sheet I have the max output rating is 1.5 amps. I'm going to say way too small for any refrigerator compressor's direct power. Maybe you can put this in another part of the circuit with lower current, like a relay or thermostat, that triggers the compressor to start. That is the way these are typically installed on air conditioners, where you put this in the low power circuit that closes the relay that starts the compressor. So if the relay circuit is delayed/open, then the compressor wont start until this device closes that circuit.

What is a good delay setting for an outside A/C unit?

The answer of 5 minutes is not necessarily right. I can guess that's the setting used by pros to avoid callbacks.Assuming the problem is tripping of the Condenser breaker, then the solution is to allow the head - high side - pressure to bleed down enough to equalize the system. Setting for 5 minutes, when 30 seconds to one minute would be enough, can be very much a nuisance. If the circulation blower runs immediately on restoration of house power, then 1 minute should be enough. Here's a way to tell without gauges - go to the inside coil location, with condenser running, listen to the hiss or gurgle where the tubing goes into the housing. Shut the power off to the furnace or air handler and see how long it takes for the sound to stop. Add 15 - 30 seconds to that and that is where to set the timer. The setting of course will not be all that accurate.Test by turning off the same furnace / air handler switch or breaker for 3 -5 seconds, then on, to see if the condenser breaker still trips.