Should no-frost fan always work?
The fan in the fridge probably doesn't have anything to do with the frost-free features (or the compressor, for that matter). Most fridges only have chilling components in the freezer and then circulate cold air into the fridge portion with a fan, according to a thermostat in the fridge.

If your freezer is frosting up I would look at the defrost heater first. It should be located behind a panel at the back of the freezer, near the evaporator coils.

It's also possible you have a defective thermostat that is not calling for the fan to circulate air into the fridge as it should.