I have used gasoline in a pump sprayer like and old 409 sprayer, and carb cleaner on a lower engine goopy greasy mess. I had to use a screwdriver and a wire brush for that deep cleaning action while wearing several layers of vinyl gloves . Used cardboard newspaper and rags to catch the drippings. Had the hose turned on and ready just in case. This is not good for rubber parts though. I let the drippings evaporate for several days in the hot sun.

Hair??? Geese, that must have been nasty! My wife came home saying she thought she ran over a squirrel but didn't see it out of the rear view mirror. I said if you don't kill them they usually cling to the underside of the wheel well until the car stops, and she ran inside.

To clean the underside, I would use a solvent based spray on engine cleaner (Gunk is one) then take it to a DIY car wash and use the hot wash. Let the exhaust cool down a little first. This is a lay down and get dirty yourself type job. [With all the hair, use the Head and Shoulders setting first. :o)]
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Tom F