I'm surprised no one responded to this. Sorry, I don't come on here much any more.
The Cooling Fan should always come on immediately when the a/c is activated. It is to keep the condensor cool.
I've noticed most of the time that when the low coolant light it on in these cars, it's because the coolant level sensor float is stuck. I tend to smack the side of the tank and the light goes off. If you have a bunch of gunk in your expansion tank, you may want to consider removing the tank, cleaning it out, and doing a coolant flush. However, it is possible that the sensor is bad.
Another sure fire way to test a fan is to pull the connector off and run 2 speaker wires to the battery *one for power, one for ground*. The fan will kick on (obviously) if the motor is good.
As far as the A/C system goes; perhaps the relay itself is bad. Sometimes you can see a burn mark on one of the pins, but an internal issue would obviously not be visible. Try going to a J/y and getting a relay. I don't know how much one is at Chevy, but you could give them a call as well. With GM, there are several functions that share the same relay. Match the part numbers. A great example; just about all the relays in a saturn are the exact same. I don't think there is one that is different.