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Here are tips for pruning once blooming and repeat blooming climbers.

https://ucanr.edu/sites/UC_Master_Gardeners/files/138925.pdf

This article should help determine when to plant roses. You can help keep garden disease free by learning the cultural needs of your plants and providing them; by buying plants that are disease resistant and being vigilant to what is going on with your plants. When necessary, take immediate action.

https://www.gardeningknowhow.com/ornamental/flowers/roses/planting-rose-bushes-in-the-fall.htm

I am not familiar with the term "pop up" rose and the internet isn't helping. If you want, repost with the formal name of the rose or what is on the plant tag. Roses are pruned in spring in NJ and most other places. You know it is pruning time when the leaf buds on the canes are swelling or just starting to open. This often coincides with forsythia blooming. Pruning stimulates new growth which is not the signal you want to send to the plant now. New growth will be tender and killed by winter. Fertilizing is saved for spring. Nitrogen moves through the soil and will largely be gone if you fertilize now. Fertilizer also sends the wrong message: your plant needs to harden off and prepare for winter. Cutting back to 6 inches isn't necessary. For now, just prune out dead, diseased and damaged canes. The following link goes to a list of articles on rose care including winterizing. https://www.gardeningknowhow.com/search?q=rose%20care

You can remove the rose hips for many uses or leave them on the bushes for the birds. I do not prune rose bushes down to the ground unless I am sure the canes have died back that far. For my winter pruning I prune the rosebushes down a bit and thin them out once some good hard freezes have hit them so they have gone dormant. This light pruning helps prevent the winter winds from whipping the canes and damaging them. It also thins out the growth to help prevent the heavy snows from causing crushing damage to the canes. Here is a link to an article I wrote on pruning for you: https://www.gardeningknowhow.com/ornamental/flowers/roses/pruning-roses.htm

Here are some tips for transplanting. Usually spring is the best time to move roses, but since you are moving, I think I would go ahead and plant it at your son's house until you are ready for it.

https://www.gardeningknowhow.com/ornamental/flowers/roses/transplanting-roses.htm

The root stock for grafting you are looking for is likely called Fortuniana. It works well for rosebushes in your area. I would look for Floribunda roses and check their growth habits. You will want the growth habit to be more bushy and medium height rather than tall and spreading I would think. You could also get some own root miniature rose bushes for the containers, they can be very eye-catching. Also contact the folks at this website: https://tamparosesociety.com/ They can give you some great local help on roses they have found to do well.