A gauge stuck in the cold position usually means an open in the circuit between the gauge and the temperature sender. Bad sender, connector dislodged, or corroded connector.
The easy way to check this is to run a small jumper wire from the sender connector to metal anywhere on the engine or chassis and turn the key to the RUN position. The gauge needle should go all the way to hot in a few seconds. If it does the sender is faulty.
The sender is also an easy, cheap repair. There’s always the possibility of a faulty dashboard gauge the the odds of that are very slim so I would shuffle that scenario to the back burner for now.
Do not confuse the gauge temperature sender with the engine coolant temp sensor (part of the engine management) or the temperature sensor which activates the radiator fan.
As to damage, this won’t hurt anything unless the car has overheated and it was not caught for whatever reason.