I mix up a bucket with very warm water and Dawn dish soap, Wet the engine bay down first with a light spray of water, then use a wash mitt and the Dawn mixture to suds up the engine bay, then rinse it off with a light spray, and blot it dry with a few old towels. The dawn cuts grease and oil quite well, but if you get it on the body of the car it will also remove every speck of wax. In fact, I use it for stripping wax when I'm doing a full detail on the paint. As far as water causing problems, a light spray isn't a problem, but NEVER use a pressure washer or coin-op high pressure on the engine.
And yes, you can go crazy and detail the entire engine compartment: