It's never a mistake to cut off dead and ugly growth, but this rose might not be the best choice for your purposes. Autumn Sunset is hardy to zone 5, but you're on the borderline, so the roots may be hardy, but the canes are likely to die back to the ground in a cold winter. It's also technically a shrub rather than a climber. It will climb to about 12 feet in the warm zones, but it isn't likely to do that in your zone. Even if it never climbs your arbor, though, it is a lovely rose and your new growth looks quite healthy.
