The cooling system is meant to control the temperature of the engine. The key word there is "control". An engine wants to run around 195F. This varies depending on engine and manufacturer.
The whole thing with "control" is that if the temperature is too high the engine will give heat away. If the temperature is too low, this is the key part, the engine will hold the heat in.
The engine does this in a couple of different ways. The first is the thermostat. The thermostat is a temperature sensitive valve. When the temperature preset of the thermostat is met it opens and allows coolant flow into the radiator. If the temperature drops the thermostat will close again. The next is the radiator. When hot coolant enters the radiator, the radiator gives the heat off to the air. If the radiator alone can not shed the heat the radiator fan kicks in to help.
The heater core in this setup is an outsider. The heater core steals heat from the engine and the engine can't control this. This can disrupt the process, kind of.
When the engine is idling it does not make all that much heat when compared to under load. Couple that with freezing temperature outside and the heat on full blast and the engine can't get up to temperature. This intern keeps the thermostat closed does not allow hot coolant to enter the radiator. This is normal operation. The engine is trying to keep in the heat until it can reach the desired temperature.
Shut off the heat. Run the engine at 2000rpm. This simulates more of a load condition. This will cause the engine temperature to rise, the thermostat to open and the radiator to get hot.
The reserve tank comes in one of two types; overflow or degass. The overflow tank just catches extra coolant when there is extra and when the cooling system gets in a vacuum it will suck some out of the overflow. The degass tank is part of the cooling system and gives the coolant a place to expand. This keeps all the coolant in the cooling system (meaning no overflow). If the degass tank overflows then something went wrong. Neither style of tank is required to get hot. They sometimes do. They both should be filled to withing the required lines on the side of the tank.