The water continues to run into and out of the water heater.
There is just no flame to heat it up.
Yes
If the pilot light is on, the heater should come on by itself. If the pilot light is off then it has to be lit. If it will not stay on, the thermocouple is bad.
Have you turned the control from Pilot to On? Have you increased the temperature setting?
Gas water heater has an orifice for main burner and for pilot light, if not electronic ignition.
The pilot setting will heat the water a few degrees. A gas water heater will have a thermal rating in BTU of 30,000 to 80,000. The pilot light by itself will have about 1,000 BTU. Operating overnight with no water movement and a well-insulated tank, you will observe a temperature rise but certainly not enough for clothes washing, dishwashers, or showers.
Pilot light alone will not produce enough heat to damage the tank, but the main burner will destroy the tank and possibly start a fire. Why would anyone want to leave the pilot light on an empty tank?
small amount
THERMOCOUPLE or defective gas valve
Same as any hot water heater. Turn the gas valve to off.
You don't. If the heater has an electronic spark, then it doesn't have a standing pilot to light. The spark lights the pilot when there is a call for heat, which in turn lights the burners. If the heater is on and the pump is RUNNING, the heater should light automatically when the water temperature is lower than the heat setting.
is the lines on the thermal coupler and pilot light for a hot water heater supposed to get hot
No, it is not dangerous. (Unless you are working with a water heater that is 20+ years old.) In any modern hot water heater, if the pilot light goes out a fail-safe will go off - preventing gas from leaking out. Also, there is a sensor so if there is not enough air-flow or oxygen with the water heater, it will shut off the gas.