How do I make a rain garden on a steep washed out area of the north side shaded part of the house?

You could terrace the slope and plant a rain garden:

http://www.pondlady.com/Articles/raingarden.html

Once the area is terraced, you can plant on the flat spots. You could add steps made from the same rocks you terraced with. I would use a flat veneer rock like Tennessee Fieldstone.

No, the trees will slide down hill. Start at the bottom of the hill, and lay rock as you come up the hill. The rock will prevent the hill side from washing away. Then plant something like Ivy in between the rock to form a matt over the rock and help hold everything in place.