You definitely have a problem. The only thing that will completely stabilize your slope is clump forming grasses - they have roots that go at least 15 feet down, virtually everything else has surface roots. And many clumping grasses will tolerate clay soil and full sun. But they don't give a groomed garden look. They can look neat and tidy but that involves a lot of work on your part, and slopes are hard to work on. If this was my problem I would aim for a wildflower meadow look with plenty of tall grasses to try to keep things stable and I would add in the daylilies, black eyed susans and such to keep something blooming all summer long. I have seen slopes covered in Verbena so some people make that work. It always dies out after a few years in my yard. Some forms of it are amazingly winter hardy and pretty much bloom year round. I have seen slopes covered in blue rug junipers also. I love the easy maintenance but you have to remove every speck of grass or weeds because they can overpower the shrubs and weeding on a steep slope is difficult. Also, any shrub is going to need a lot of watering to get it established and if this summer is a dry one, that will be hard to do. If the clay is holding up and not washing downhill you may not need to worry about erosion control at first. It all depends on just how steep your slope is. If you live in a neighborhood where a wildflower meadow out front won't go over well you could choose one type of plant and cover the entire slope with it. That mass of uniformity will look managed and planned and hopefully solve your problem and depending on what you grow could be virtually maintenance free.

On my hill I use Daylily (Hemerocallis fulva)and Nepta"Six Hills Giant" together. I like this daylily because it spreads by runners where as the hybrid daylilies are clump forming. Both the daylily and the nepta put on a great show.

I'm not sure what kind of tree that is. With some trees, having the roots that way is normal and healthy. Some trees suffer if their surface roots are buried so research that before laying on a lot of soil. The water coming off of the pavement has a lot of velocity. Porous pavement is very green but expensive. Conceptually, look up things like french drains and water bars. You can also surf Erosion Control magazine for free on the web. Also check websites for Ohio's department of transportation and department of environmental protection (or whatever the equivalent departments are in Ohio). Here is a link that might be useful: Erosion Control magazine

I have sumac Low Grow, and it is doing very well in part Sun which surprises me. It is not particularly low maintenance because of the taller deciduous trees above it. The fallen leaves smother the sumac in the fall and come spring those leaves are a great, sodden, heavy mess. This is a steep hill and raking even a little is not easy. OTOH, it is a good spreader and fall color, even in part sun is excellent. I have large patches of ground cover geraniums, macrrorizum Spessart and the magenta species which are steady growers and need little water and very dense after a few years.