You possibly need a new 'thermocouple' -this is a device that sends a signal to the regulator to turn on. They commonly fail, not a big job for a gas technician. -Call one to repair it, about a 1 hour job.
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.
Yes
Same as any hot water heater. Turn the gas valve to off.
The water continues to run into and out of the water heater. There is just no flame to heat it up.
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.
No because water heaters have a safety valves and when the pilot light goes out the safety valve shuts off the gas
Thermocouple is most likely the problem but it also can be wind and draft which blows out the pilot light and very rare it can be you have a air pocket in your line or sometimes condensation can drip on the pilot. If you have a flame when starting the pilot light as you are pushing down on a non forced air gas water heater and the pilot goes out after you release the pilot handle then 90 out of a 100 times its your Thermocouple. If you have a forced air tank type gas water heater it can be a lot more sophisticated and I recommend you read your manual trouble shoot guide and follow those steps for YOUR specific water heater.
It is most likely the thermocouple that is bad. That is what keeps the pilot light on. The thermocouple is the small tube that sits in the flame of the pilot and runs to the control valve.
IT turns the heater on and off!
You set a heater to a certain temperature and when it reaches that it turns off.
No it shouldn't, There is a sensor that turns off the gas if the pilot flame goes out, The only way it would leak gas is if it as a faulty sensor, or a peice of depree gets in the gas line
If it is windy outside, you may be getting just enough of a breeze to blow out the pilot light -even though the covers are shut. My heater is in the garage, and even w/ the garage door shut and the covers on the heater shut, very windy weather can still cause enough of a draft to blow out the pilot. You could try to find something to block drafts around the heater such as a hot water heater blanket.