When I shut off the cold water intake to the hot water tank, cold water started flowing from my bathroom tub faucet. HOW AND WHY?

This is mainly just due to air pressure. If you shut off the air pressure in your plumbing circuit in the home, water may flow out of some of the pipes for a little while until the pressure is all run through the pipes. It should only happen for a minute or two at most and then the water should stop as the pressure runs out.