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Most rosebushes are grafted rosesbushes. The top part being the rose that we bought and desired in our garden or rose bed. The bottom part is called the root stock, it is from an entirely different rosebush that is far more hardy. Thus the rosebush is able to survive better in most climates. When the top part that we desired dies for some reason, the root stock sometimes takes over and will send up its new shoots or canes. The bloom we have then is the bloom of the rootstock and the rose we originally desired is gone. Here is a link to an article about grafted and Own Root roses for you: https://www.gardeningknowhow.com/ornamental/flowers/roses/own-root-roses-grafted-roses.htm

I am not sure where you are located. However, here are some fine rose foods that I have used and like. Some may be available in my area only.

Mile Hi Rose Food, Locally made product in Colorado.
Gro-Rich Rose and Perennials Food - Made by Richlawn and available at some Home Depot locations.
Vigoro Rose Food, also available at many Home Depots.

Look for a rose food that has some alfalfa meal and or kelp meal in it. The roses love those two meals alot. The organic nature of the top two foods I listed help build the soils as well as feed the rosebushes.

Moving a rose bush that old is tricky. They are very, very susceptible to transplant shock and going in a car for that long will not help the matter. I am just telling you so that you are aware that it may not make it no matter how careful you are..but these articles will help you do the very best you can for it during the move: https://www.gardeningknowhow.com/ornamental/flowers/roses/transplanting-roses.htm
https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm

Also, you will want to start cuttings, just in case it does not make it. That way you will not completely lose such a valuable heirloom. This article will help with that: https://www.gardeningknowhow.com/ornamental/flowers/roses/roses-from-cuttings.htm

Sounds like your mini rosebush shot up what is called a Blind Shoot. Some do that from time to time. I just prune them back to more the height of the rest of the rosebush. Then see if it sends out some new growth from the junction of a leaf set and the cane. Those then should go ahead and form buds to bloom. If they do not, then you will need to prune that one cane out as far down as possible.

I have read that using an organic (earth friendly) type liquid soap mixed with water, 1 tablespoon to a quart of water, then sprayed on them will get rid of them. I have also had times when just spraying the bushes down with a fairly hard spraying of water a couple times a day for 4 or 5 day days will send them packing. If you are under heavy attack and they will not leave, then you may want to buy some Bug B Gon Max that is in a hose end sprayer and spray the roses down with that. Usually one or two sprayings will get rid of them. There are some insecticides by Safer and one that is called Eco Sense that work pretty well too.