Deadheading is just snapping off the spent bloom with your fingers. The old "cut to a 5-leaflet leaf" rule is unnecessary; modern research shows simple removal works fine.
The best way to preserve your mother's rose is by taking softwood cuttings in early summer. A detailed, reliable guide is available at https://www.rdrop.com/~paul/hulse.html. Take several cuttings, as not all will root.
Moving a 25-year-old rose is risky. If you must, do it when dormant, cut back by half, dig a massive root ball, and provide relentless aftercare.
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