The valve opens up when it reaches its predetermined temperature - usually 82 degrees celcius and closes at anything below it - the engine itself produces heat and the radiator takes it away - this thermostat regulates the engine coolant temperature by stopping flow of coolant into the radiator - instead it recirculates it in the engine itself (what we call bypass).
The radiator fan is controlled by the thermoswitch which is usually installed after the thermostat valve or on the farther radiator tank - and is tripped by the hot coolant - once the coolant cools off from the radiator the switch is retracted and the fan stops.
As I said both items are separate.
As your thinking that the gauge thermosensor is faulty and not having stable resistance is spot on - but Ive encountered similar problems with cars having a bad wire - usually corroded right at the thermister connector or the body wiring connector.