Can you hose off an engine to cool it off without damaging the engine?

I wouldn't. You may crack something such as the exhaust manifolds. Better to let it cool down on it's own. Also, it is usually best to let the engine run to cool itself down. If it's not running the coolant can't circulate and the fan may not function. Unless there is no coolant, then you're s.o.l. and might wanna pray a little.