What kind of tree is it?

Live Oak

That's what I thought. If you put anything loose like gravel, you're going to have to deal with leaves in the gravel and seedlings everywhere. Those things are prolific buggers. What would you think about pavestone? I don't know how long it will be before the roots cause them to shift but at least they can be reset, unlike cement.

I'd ditch the idea of stepping stones and gravel as it would be temporary and messy. It's not that difficult--or expensive--to form up and place a small amount of concrete and if you used that, you'd have something durable and lasting. Nothing against concrete pavers on a well tamped base so that's another option, too.

Per usual, I'm submitting my suggestions in pictorial form so they're mostly self-explanatory. The returning branches ... I'm showing how they're going to be at the end of summer. Better to cut off some low-hanging limbs sooner than later (since you know it's coming) and get rid of some of the "gloom effect" that the heavy shade produces. For a groundcover, you might look into Calyptocarpus vialis, which goes by several common names. Plants are what hold soil and keep it from washing, so if you used groundcover below the tree in the shady area, you could eliminate the soil washing problem.

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Thanks Yardvaark! This is very helpful. When we go to plant the Calyptocarpus vialis, I am assuming that we should bring in some good dirt and start with that? Then space the plantings every few feet where we can under the tree and they will grow together?

I'm sure your dirt is good enough. Just kill weeds. Mulch. It will grow together.

You might check to see if removing and/or covering roots with concrete or pavers will harm the tree.

Fall is the best time if you must really disturb the root zone. Pavers plopped on the soil can't harm the tree unless you paver the whole yard. Concrete over more than 1/4 or so of the root zone will probably be a thing.

Suggest you just apply lots of mulch to the entire area for now. That will eliminate the muddy soil. The small patches of existing grass should either be killed or covered in cardboard before mulching. A few large containers with colorful plants grouped together would provide interest.

You don't state where you are. For groundcover choices, see what others in your area have planted in similar conditions.