It is certainly achievable in your zone since I grow both roses and clematis, but not in a north-facing site. Both clematis and roses need sun to bloom, IME a minimum of 5 hours, and more sun will give more blooms. So I would plan to put your roses and clematis in sunny spots and find something that is more shade tolerant for your north-facing garage.

IME roses will do fine on a trellis like the one you have in the photo, but will need to be fastened to the trellis since they don't cling on their own. Most clematis will cling on their own, but need something pencil thickness or less since they climb by wrapping their leaf stems, which are relatively short, around the support. So either they would need to wrap around the rose stems (so you would have to find a rose with hardy enough canes) or you would need to add something such as wire mesh (like hog fencing) or strings to allow the clematis to climb. I most often choose metal supports since they are easier to create and maintain with thin enough bars.

You don't mention where you are in zone 5. In my NH gardens (and where I grew up in OH) Japanese beetles are a problem for roses and I am not willing to use pesticides since they also kill other pollinators such as bees. So I choose roses that don't bloom midsummer when the beetles are at their destructive worst. There are a fair number of older varieties of climbing roses that bloom just once in late spring/early summer in our zone, before the J. beetles emerge. I also grow one old Bourbon rose called Zephirine Drouhin which blooms for me in the spring and fall, but takes a break midsummer. It has bright pink roses and a lovely scent and is easy to work with since it is almost thornless which is handy when pruning both the rose and its accompanying clematis. I've mostly seen it rated to zone 6, but it has survived several winters here including one that got to -23. It does sometimes die back a fair amount but regrows to about 7' each year. There are several roses developed by the Morden Agricultural Station in Canada that are quite hardy, including some shorter climbers (6' or so) but they tend to overlap bloom with the J. beetles, so I haven't been as happy with them. If you don't have the beetles you should be fine. You can ask for additional input on the rose forums.

How large to make a trellis will depend on how large your plants are likely to get, both the rose and the clematis. You want to choose a clematis that isn't so large and vigorous that it will overwhelm the rose, so I'd avoid the most enthusiastic clematis such as Betty Corning or sweet autumn. There are a lot of really lovely deep purple clematis such as Viola, Jackmanii, Etoile Violette, and Polish Spirit, all of which bloom midsummer. In general the summer bloomers (hard prune AKA type 3) are a bit easier than the large-flowered spring bloomers (type 2) but currently most of my spring blooming roses only overlap with my summer blooming clematis for a short time. Of those I listed, I only have experience with Viola, and it would be fine with a rose IME. If you are interested in other color clematis, I'd look at various online nurseries to see what is available. I've ordered from Brushwood, Joy Creek, Bluestone, Hummingbird Farm, and Silver Star Vinery. Each has a different selection, and some send quarts or smaller while others send gallons. In a more urban area than mine you might have a larger selection available at local nurseries, but IME most of what is available here are spring bloomers since that is when most folks shop for plants, and blooming plants usually sell better than bare vines.

NHBabs, you have been so incredibly helpful and informative for this and my hydrangea questions!

We are in northwest indiana about 10 mins off lake michigan, zone 5. So when we get cold, we get cold and snowy. I think the roses may be too troublesome for me. I recall our knockout roses getting demolished by the beetles in Chicagoland before we moved out to NW Indiana, so I'm shying away from them.

Do you think any clematis can grow on the North aspect of our house. We are turned a bit toward the west, so some of the garage area may get some afternoon and evening sun. Hard to know for sure because it's new construction and we aren't living there yet.

Clematis and hydrangea are two of my favorite plants!

I'd wait and watch that wall to see how much sun the area gets this growing season. Check several times over the first half of the season since as the sun moves, you may find that there are parts of the growing season when it gets little sun, but it looks like it may get reasonable western sun during a good chunk of midsummer. I might go out there now and every couple of weeks throughout the spring since by early April it will be approximately the same as late August as far as sun exposure and by midJune you have the longest day. You could then decide on a clematis if appropriate and order for early fall planting. There are some clematis that can take less sun but I don't know of any that can get less than 4 hours and still bloom well.

As long as you don't mind having roses that only bloom once in spring, they should be fine as long as they get enough sun. Knockouts bloom all season which is why they get eaten by the beetles.

Another combination I like (since you are considering planting both plant types) is Clematis growing up into H. paniculata like Limelight. Here is H. paniculata 'Quickfire' with Clematis 'Little Bas' growing up into it.

I plant the clematis at the ultimate dripline of the Hydrangea and then give it a bit of support until the Hydrangea reaches size.

Hayley grows well on the north side of our house. I don't grow roses so have tried other clematis to grow with Harley but they haven't done as well.