I just got my new Igloo 7.1 freezer and the compressor light was on. Now it's off. Does the compressor light cycle off and on for the freezer?

Sounds like relay on compressor is bad or worse the compressor is bad. Return it

Compressor may be bad. Also could be the relay that plugs into side of compressor that’s bad. Inexpensive, easy to replace part if it’s the relay.

A loud noise when turning on an Igloo portable ice maker could indicate a problem with the compressor or a malfunctioning fan. It is recommended to unplug the appliance and have it inspected and repaired by a qualified technician.

What if the red light is on and the green light is off what's the problem

Great ? but need more info
1 is compressor running or fans running can be compressor has stopped running and need new start relay

door seal not sealing
compressor not working
could be a start/run capacitor , low gas or a seized compressor
cal a technician for a proper diagnosis and quote

Could be a loose line or etc vibrating against something else or the compressor motor could be shot.

Defrost heater.

It should stop the compressor and turn on a heating element to speed up defrosting.

The compressor may be frozen. Turn off the fridge for 6-8 hours, then turn it back on. Don't turn the temp to the coldest setting. If it will still not get cold, the compressor is probably bad.