That's a good set of photos for an identification. The foliage appears to be that of a hybrid rugosa. I have no experience with these because I've never grown rugosas in my garden, but I did a search for hybrid rugosas of the same color and petal count as yours and found that the buds on most of them are a deep pink or are multi-colored. The only buds I found that match yours are on Conrad Ferdinand Meyer, an 1897 hybrid rugosa.
Rose (Rosa 'Conrad Ferdinand Meyer')
Hybrid rugosas are offered for sale by all of the own-root rose nurseries, so they must be relatively easy to propagate by cuttings. I grow cuttings in 1-gallon containers. If you want to do that, choose tips with buds on them. They are at their peak for reproduction when they display the energy to produce buds. The cuttings should be about 8-10 inches. Cut off the buds and any flowers at the top of the cutting, remove the leaves from the bottom 2-4 inches, and plunge the cutting into a well-draining planting mix, burying the leafless bottom 2-4 inches. If you wish, you can apply powdered rooting hormone to the bottom of the cutting. That can help to keep a cutting alive. Keep the planting mix moist, but not wet, and protect the cutting by keeping it in dappled shade until it produces new leaves of its own, at which point you can start moving the container to acclimate the rose to gradually sunnier locations.
If you want to move the entire plant, that can also be done. Water the soil around the rose thoroughly, dig out the roots, prune the top of the rose bush back to match the length of the roots, and plant the bush in the new spot, making sure to water the plant daily until it has adapted to its new location. It would be best to rig up some contraption to shade the plant during the hottest part of the day until it had adapted to the new location.
If your rose is a hybrid rugosa, there's another propagation option. Rugosas send out runners, and it's likely that your branch coming up a short distance away from the main plant is a runner with roots on it. You can try cutting that runner off the main plant and planting it separately, either in a container or in the new location. Again, the key to keeping a transplanted rose alive is daily watering for the initial period and protection from intense sunlight until the rose is happy in its new spot.
Here's a tip on the general care and maintenance of your rose. Don't ever spray it with pesticides, fungicides, or liquid fertilizers. Rugosas hate being sprayed and will react negatively to any kind of spray.
