Why is cold water coming from the hot part of the faucet?
When he closed the valve on the heater cold water inlet, the pressure should have dropped off and the water ceased to flow. The hot water circuit is supposed to be fed from the cold water circuit only through the shutoff valve on the water heater.

Continued flow of water in the system indicates there is a cross-connect somewhere in the system between the hot water circuit and the cold water circuit. 

Usually this happens where you have one of those instant hot water circulator pumps. They monitor the temperature in the hot circuit at the remote location where you want quicker delivery of hot water and push the cool water from the hot circuit into the cold circuit till the hot water circuit reaches a set temperature. If the pressure in the hot water circuit drops, there should be a check valve that prevents cold water from pushing back through the hot water circuit.

Another potential cross-connect is where you have a shower with a balance valve. It works by first turning the cold water on full and then slowly turning on the hot water. If the plumber had the shower turned on as one of the hot water faucets, it could backfeed cold water into the hot water system as there's no pressure in the hot water system to prevent it.

The solenoid valves on the washing machine could also do this under rather odd circumstances, the easiest way of eliminating that is to just operate the water cutoff valve. In modern installations it's a dual ball, single lever valve that shuts both hot and cold off to the washing machine.

All other cross connect type water flow would require faucets to have both hot and cold water on simultaneously, feeding back some cold water into the unpressurized hot circuit, which if he only ran hot water through them, wouldn't occur.

The final issue is a plumbing misconfiguration somewhere, you probably would have already noticed that.