Does anyone know what type of grass this is? I live in Central NC and the grass is greener in the summer. There is also moss mixed in with the grass.

I would probably transplant the clump of ornamental grass. It is too tall for its location and consuming the more attractive ornamental tree. I, too, advise increasing the size of the planting bed beside your driveway, adding curves that will permit you to plant appropriate perennials as well as annuals as the season permits. The grass is a nice plant in the wrong place, I think. It will look better in a less obvious spot in your yard.

What is creating the shade..is it a tree that brings something to the surrounding soil that does not agree with the grass....for example oak trees create an acidic soil unfriendly to grasses..or maybe not enough sun..consider a different ground cover type plant for that area or shade plants if sun is sparse...many trees do not like as much water as grass requires...perhaps just mulch the area and put a nice bench around the tree

Yes, it looks like heavy clay soil. Try amending the soil so grass will grow there. I don't think it has anything do to with the snow deposits, I think you graded in crappy soil and now nothing will grow. Another solution, install a raised bed for vegetables or flowers