I cannot be much help here as I cannot find any tech info on this unit.

I am assuming that the actual part you want to control is the compressor.
Try to get a wiring diagram of the unit.
You need to know what device measures the temperature (thermistor) and whether is is a positive of negative coefficient device.
For instance a negative device gives a lower resistance as the temperature rises. So a high resistance means that the temperature is low and the compressor does not start.

Once you know this you could use your timer to run a 120 volt relay.
The relay contacts are then wired into the thermistor circuit.

For instance if the thermistor is negative coefficient the contacts are wired in one of the wires going to the thermistor (wired in series). With the relay off the unit sees an open (infinite ohms) as the temperature so it thinks that the room is cold so does not turn on the compressor. When they close for the 20 minutes it sees the actual temperature/thermistor and will start the compressor if the measured temperature is higher than the set point.

The above will not work if the unit has a self diagnostic which senses an open thermistor. In this case a you would need a relay with normally open (NO) and normally closed (NC) contacts. A resistor is then wired across (parallel) the normally open contacts so the unit sees this as the thermistor resistance and will not start but also will not cause an error.
The resistor of course has to be of the correct value.

If it is positive coefficient then you need a relay with NO and NC contacts and when you want the unit to stay off a resistor of the correct resistance is wired in so the control thinks the ambient temperature is below set point.

I hope the above makes sense and is useful to you.

P.S.: GE likes to keep everything a secret so getting the required info may be a pain.

Cannot say as I do not have the required info.

It should be low voltage so should not cause a problem but I cannot be 100% sure.