Water – How full does a home water heater get

Yes, it's a thermostat, and there is usually a dial on the side of the tank, near the bottom, to adjust the temperature. More than likely, the water feels colder because it is traveling through uninsulated pipes and is losing much of its heat before it reaches the faucet. You can turn up the temperature of the tank, insulate the pipes (they make foam insulation that just slips over the pipe, just be sure to measure the diameter of your pipes so you get the right one), or both.

You should also consider flushing out any sediment from the bottom of the tank if it's been a while since you last did this. All you need to do is hook up a garden hose to the drain at the bottom of the tank (make sure it can take the warm water), run the other end to a floor drain or utility sink, and open the drain on the bottom of the heater. Make sure the water runs clear for a minute or two before closing the drain. To save on the gas, you can set the thermostat to only run the pilot light while you flush the sediment.

Any leak large enough to stop the water from filling the heater tank would have to be so large as to basically be an open ended line. It would be impossible not to notice the flooding caused by a leak this large. A smaller leak would not stop the heater from filling and operating and you would have some hot/warm water delivery. I am not saying you don't have a leak, but I strongly suspect an electrical issue. It could be related to power supply, or it could be related to damage caused by improper heater installation or operation. Call a reputable professional plumber please.