I'd love some photos that have more even lighting such as on a cloudy day and if possible one from overhead such as a second floor window. I am having a tough time distinguishing things with this sun/shadow lighting. If you can add where generally your zone 7 is, that will help since there's a fair variety in soils and general climate between various zone 7's (OK vs WA vs NJ for instance.)

1. Lonicera sempervirens is long-blooming and will be largely evergreen in your zone. It will climb a trellis but may need pruning since it can get largish. If you can give up on evergreen, there are quite a number of lovely clematis, but I don't think that there are any reliable zone 7 evergreen clematis. The hard prune ones could be pruned back in very late fall after hard freeze and you could just enjoy the bare ornamental trellis, perhaps strung with lights for the winter. Gelsemium sempervirens 'Margarita' is not a climber, but is sort of viney and blooms quite early - perhaps April. It does better draping but could probably be guided up a support and clipped.

3. Here's a thread with a bunch of suggestions for evergreen groundcovers good for New England zone 6 for your sunnier areas to plant around the shrubs. For shadier areas, look at Asarum/ginger. I grow European ginger, plus there are other gingers with marbled leaves that you might grow that I can't due to hardiness issues. I know that Plant Delights Nursery has quite a lovely selection. Hosta, Iris cristata, Astilbe, and Pulmonaria all are happy in at least some shade and combine blooms and foliage, though the Iris is just blades. Epimediums would be another early shade tolerant bloomer for you, and some are evergreen. Hellebores are another. I use all these plants around the feet of shrubs, planting them small and then letting them ramble or expand to fill space. I often use gold foliage to create a pop of color in semi-shady areas, and it lasts longer than most perennial flowers.

There are relatively narrow evergreen shrubs, but I wouldn't know what to recommend for your zone. You could see what might work by asking on the shrub forum.

Pretty much the things I've recommended do fine in New England in my well-drained acid soil, so if yours is too different, you will need to research if they will do well for your growing conditions.

It looks like you might be in an urban area and dealing with some challenges in terms of space and sunlight. I can definitely sympathize, and have been working on making my urban garden more cottagey for 5 or so years now. Your courtyard is so charming and the hydrangeas will be so lovely and impactful when they mature. In which direction is your garden facing, and have you done a thorough assessment of full sun vs medium sun vs shade areas? I would recommend doing a mindful study of your light variations around the summer solstice if you haven't already. In my garden, some spots get practically full sun, while others just a few feet away only get two or so hours a day due to hard shadows from roofs/buildings etc.

As for your second question, it's a bit hard to tell scale from your photo, but I'm not sure about a large plant to replace your hanging basket. Your weeping birch probably has a lot more growth to put on, and I would worry that something planted that close would compete too much for sun and root space.

Babs has some good suggestions for evergreen groundcovers and I second hellebores, a personal favorite. Pachysandra is another great utilitarian evergreen groundcover, but somewhat boring.

You can do a lot with annuals that like shade: coleus, begonias, impatiens, and nicotiana would be some to try, and they provide some color for practically the entire growing season. I'm visualizing an array of different annuals (in pots and in ground) along your lefthand border, around and between the hydrangeas until they fill in. Container plantings are fabulous for small spaces and they are mobile, so you can move them at will. Take a look at ferns as well.

Here is my urban cottage garden last summer.