Answer: I'd say you have some very particular requirements for your tree, but there are several that fit within the criteria. Some trees are deeper rooted than others, but a tree planted in a lawn where moisture will be frequently applied will have an abundance of roots growing close to the surface of the soil.
Fruitless mulberry is a fast-growing, wide spreading tree. (The female, or fruit-bearing species can make quite a mess!) Catalpa grows fast and makes an excellent shade tree. If you're looking for fall color, Sweet Gum, sometimes called Liquidambar, would be a good choice as well.
Hope at least one of the above suggestions appeals to you!
Q&A Library Searching Tips