This sounds as if the 12 volt to the board is missing. When you turn on a light you're getting the 12 volts to the board.
Locate the 12 volt input on the board and stick an external 12 volts there and see if the refrigerator works.
On AC voltage the board still needs 12 volts for operation. The AC is for the heating elements but doesn't supply power to the board.
I can't comment to the furnance deal but it smells like a wiring problem of some sort.
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