Yeah. But a good degreaser works better than the so called "engine cleaner" found at auto parts stores.

Yes, do NOT put cold water on a warm engine.
If you are trying to clean your engine with degreaser and a brush, you don't want to get in there when it's hot just for the fact that it's uncomfortable (unless it's 30 degrees outside and the engine is nice and warm... in which case you are a fool for trying to clean your engine in that weather) plus the degreaser will dry up before you finish cleaning everything. Do it in the morning with the engine cool, and keep the hose away from electrical stuff as much as possible. Use rags with degreaser to wipe them clean rather than spray them with degreaser and rinsing it off.

PS. MAKE SURE TO CLEAN UNDER THE HOOD TOO!!!
I've seen cars with nice clean engines, and nasty dirty hoods.

And don't forget to armor-all the hoses and plastic stuff when you are done. NICE