Well you will have to determine why it is leaking. It is just a loose fitting? try tightening it up with a wrench Is there a crack? Is the pipe white PVC (plastic) or copper or even gray metal? If it is any of these you will need to cut it out and replace that section. You might want to consider hiring a plumber.
Cold water pipes would be the sweaty ones.
It doesn't mend the pipe. The pipe will still have a leak. However, you can use liquid nitrogen to freeze the water in a pipe and create a 'freeze block' which will prevent more liquid from getting to the leak. Once the freeze block is in place, it is much easier to repair the pipe leak.
That depends upon the length and diameter of the pipe connecting to the hot water tank. Also, the greater the distance, the more hot/cold mixing will occur, resulting in even greater water loss. Most copper household plumbing pipe is 1/2" in diameter, and hot water pipes may range from 10' to 80', depending on angles and turns in the plumbing. An 80' pipe will hold about 2 gallons of water. This is water that was heated in the water heater, but upon setting in the pipe gave up all, or most, of its heat to its surroundings.
Depends on the size of the heater Commerical etc
Assuming you are talking about a home water heater, there is a pressure relief valve above the hot water line - usually on a tee or an ell above where the hot water line comes out of the heater and usually leads to the floor and under the house. The valve is designed to relieve the pressure in the water heater when the water becomes too hot. The valve may have corroded and is stuck in the open position thereby releasing the hot water. You can feel the pipe that connects to the valve = if it is hot, it is leaking water; if it is cold, the problem lies elsewhere - maybe the thermostat could be set too low.
Hot water pipe rust first as compared to the cold water pipe for the same chemical composition of the water.
Copper can rust
Cold water pipe.
Seep
leaking
If you are referring to a regular building, the cold water feed pipe goes directly to all cold faucets, toilets and hot water tank. The purpose seems obvious.
Electrical grounded to the cold water pipe, usually before the first fitting once it is through the foundation..
Vent, return circulation
If it runs continually, as you say, then it can only be the inlet water pipe or fitting, leaking.
Repairing the section of pipe that is leaking will stop a leak on a hot water line.
Because the cold makes them slightly more brittle and the water in them freezes and in doing so it expands (needing more room) and this makes the pipes bulge and crack. You discover the problem as the ice melts!
If you are referring to a regular building, the cold water feed pipe goes directly to all cold faucets, toilets and hot water tank. The purpose seems obvious.