I'm having a hard time figuring out exactly what you're saying in your original post, particularly as regards the hot circuit.
Here's what you can do for the fan though.
You can get an AC fan for pretty cheap, and then you don't have the extra step of socket->power adapter->fan. I threw one of these in my ferm chamber:
You can wire it directly to the cold control terminals on the thermostat, parallel to the compressor. Make sure to ground the chassis of the fan to the fridge with mounting screws to eliminate the chance of an unpleasant shock.
That being said, you probably want the fan to shut off when you open the lid, no? If that is the case, I would get a normally-closed relay or some solid-state switching device, wire the relay coil to the lid switch, and run the load side of the relay in parallel to the compressor on the thermostat switch. That way, the fan will only run if the lid is shut and the temperature is above the set point.