1. Is this bad for the condensing unit?
Yes, you stand the chance of freezing up the evaporator coils like a block of ice.
2. Is this a problem in the condensing unit or the air handler?
Don't know, will have to troubleshoot to find the problem. Possibly a stuck contactor on the compressor.
3. Does the air handler control the condensing unit?
No
4. Would either of the devices have an automatic shutoff that stops the device from working at a certain outdoor temperature?
It is possible (but not likely) that there is a low ambient tempature sensor that will cut out the compressor, not the air handler.
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