You should not prune any roses severely in the fall. Pruning will stimulate new growth that will be lost over winter. In your zone your winters are milder than mine but you do still have them. The roses store energy for the next spring in those canes. If you cut them off in the fall you deprive the rose of that needed energy and risk losing the rose in the spring. If you are concerned about the long canes whipping in the wind over winter stake and tie them so they won't do that.

Those canes coming up from the ground in front of the bud union are not going to be your original rose. They are the root stock coming up. Probably Dr. Huey which is a once blooming red climber. The fact that it only blooms once a year is another reason not to prune it now if you wish to keep it. Once bloomers only bloom on old, last years, wood. Which is why it did not bloom this year on those new canes. If you prune those canes off now it will not bloom next year either. Once bloomers should only be pruned immediately AFTER they have finished blooming in the spring. That way it will have the rest of the year to grow back canes to bloom on next spring.

The best time to transplant any rose is in the late winter or early spring while the rose is still dormant. If done properly the rose will hardly notice that it has been moved. Prep a big hole in the new spot first then dig out as big a root ball as possible and move it to the new site and replant. Keep it watered and wait for it to start to show new growth before adding any kind of fertilizer. It will need to replace some of the fine feeder roots and they will be very tender and could burn if fertilized too soon.

Seil:

Thanks so much for your comprehensive response to my question.

I am confident that your help will make a difference in my roses come next summer.