Seeing the picture, I am wondering what the hell the dome is about ... so the final sentence did the explaining and made me chuckle. As far as "fitting in" goes, I'm a believer in it when what one is fitting in TO, is as desirable as possible. But if there is room for improvement, then there is no reason to shirk from improving and setting the example that will inspire others. Over years, a neighborhood can completely transform for the better on account of trend-setting gentrification. In this case, the street is already and oddball mix of structures. if you improve it, you will be seen as leaders and others will examine what you do and get ideas from it. Trust me, copying is rampant.

Here's two views of a nice little formal garden down the street from me that I love walking past. (The dome shaped tree is actually a lilac!) This is, believe it or not, the 2 1/2' wide hell strip between the sidewalk and the street, on another walk.

I have to disagree with you, Emmarene, that the slope is too large for a single groundcover. A groundcover can be the LAWN where a mowed turf grass lawn is not possible. One must just pick the groundcover that works for such a size. Some "timid" groundcovers work best for tiny places. Some more aggressive groundcovers work well only when the area is large enough, considering that mainly, one is managing EDGES. An aggressive groundcover in a small area would be a constant battle, but in a large area it can be one's best friend ... making ground green when grass can't, or won't, do it. The place to begin is finding out what others in the surrounding area (town/county, etc.) are using to solve the exact problem in their yards. Then one knows it grows in the area and is probably well suited to the size & job. If one groundcover is used predominantly over all others, then it is the groundcover to use. This is not the place to "be different," trying to express one's individuality. This is especially true if there is groundcover in use in a neighbor's yard. It would be an impossible battle trying to keep two different groundcovers from mixing at a property line. If one type of groundcover flows from one yard to the other, then there is no edge to maintain at the property line, making one's list of chores less. As an example, while risking bringing the haters out of the woodwork on account of the mere mention of it, I'll say that if this property was in Atlanta, Ga., the clear cut choice of groundcover would be English ivy, as it is the one in predominant use there for hilly and shady large areas (and some surprisingly small ones, though that is not my taste.) While many froth at the mouth at its mention, it nevertheless proves to be successfully manageable by many. (For me personally when I lived there, I considered it easy to care for, a lifesaver ... more or less worthy of worship on a grand scale, as it was the most versatile, hardest working horse in the barn.) In the typical case, since it commonly flowed from neighbor to neighbor, there was not property line trimming of it. Where it met lawn, many people were happy just to let the lawnmower be the one and only edger. Where ivy grew into the lawn, the regular mowing kept it from being noticeable at any distance. That left the only edging to be done where the ivy met paving. There, it needed to be trimmed a couple times/month during the growing season. One trimming per year kept it from climbing trees if one so chose. At back alleys and buildings, its leading edge could be sprayed with herbicide (such as Round-Up) in order to keep its advance in check. Or it could be manually cut, if one preferred. (Another advantage of it is that it was extraordinarily cheap/free/easy to create massive quantities of it, if one learned some simple propagation techniques.) Insofar as dividing groundcover from lawn, one would mark out all space that is too shady for turf grass to grow, or too hilly for it to be mowed. And then apply some simple art to it in order to come up with a dividing line that looks pleasant. In this case I would leave an "L" shape of grass next to the street and the drive, where it is neither too shady or too sloped. A radius transition between the two legs of the "L" would work well around the tree. The tree would be in the groundcover section, a visually comfortable distance away from its edge. Speaking of the tree, it is well past time to remove the scruffy looking bottom limbs from its trunk. The "shade" should not be hanging down, obscuring any front portion of the house.

I like the stonework you installed. Where are you located (and/or if in the US, what USDA plant hardiness zone are you in)? Where you live will dictate which plants are hardy and/or available for your area. And do you think you want mostly low-height groundcovers for this area? Assuming you are in the US (and esp if you are in the Midwest or eastern half of the US), I would consider the following native plants for your site: *Pennsylvania sedge: great alternative to Japanese forest grass, these work well for shady sites and, when planted in masses, give a modern vibe that I think would work well with your hardscaping *Wild ginger: another native that will fill in as an attractive green groundcover— has larger leaves that look good next to hosta. Flowering natives that stay low to the ground that could also be good options in shade (most blooming in spring): trillium, Canada anemone, wild geranium, bloodroot. And if you are wanting a flowering native that gets some height, one that could be planted right next to your hornbeam is goatsbeard. My last yard was deep shade and these thrived at the base of a large spruce tree, getting white plumes of flowers in the summer. Good luck with your project!