Running the engine won't drain any more coolant than what you'd get just by opening the radiator drain. Once the water level drops below the level of the water pump, it quits pumping. Its just a centrifugal pump, not a positive displacement pump like an oil pump.
The best way to completely flush an engine is to open the radiator drain plug and also open the block drain plugs, if so equipped. The problem is the block drain plugs are usually stuck *hard* in the block.
Not a good idea. Hot spots can develop in the head very quickly and without the water cooling, warping or cracking can occur. Why not dop the return line and flush water back through the heater core or a flush tee in the core and run the engine while flushing, much safer.