Most of North American species spread by suckering, so given time you will have more plants. Sometimes a lot more plants. Suckers were probably what you were given originally. It can be pruned to shape any time during the summer. A lot of us do the major pruning on once bloomers in early spring. It isn't something to really stress over because if you goof, the rose will get over it, and you can try again. It may take a year or two, but it will work out.

Here is my revised list: Revised 'Old Garden Roses' for Small Gardens Dream List (08/21/2013) 1. Autumn Damask (Damask ancient) (gets big but I like it. I have to figure how to use it. I like this rose because of its rose history) 2. Boule de Neige (Bourbon, 1867) 3. Honoraine de Brabant (Bourbon, age unknown) (gets big but I like it. I have to figure how to use it) 4. Maggie (Bourbon found 1980) 5. Souvenir de la Malmaison (Bourbon introduced 1843) 6. Madame Cornelissen (Bourbon introduced 1860 to 1869) (Thank you Ingrid_vc for the suggestion) 7. Beauty of Rosemawr (Tea, 1903) 8. Westside Road Cream Tea (Thank you Luxrosa for the suggestion) 9. Captain Harry Stebbins (Hybrid Tea discovered 1980) 10. Devoniensis (Tea, Foster 1838) (Thank you Ingrid_vc) 11. Francis Dubreuil (Tea 1984) 12. Comtesse du Cayla (China 1902) 13. Fabvier (Hybrid China before 1929) (Thank you Luxrosa for the suggestion) (does it have a fragrance?) 14. Green Rose (China prior to 1845) (I like this humble rose) 15. Old Blush (China Hybrid, 1852) (I like this rose because of its rose history) 16. Fritz Nobis (Floribunda, Shrub 1940) (not a OGR; gets big but I like it. I have to figure how to use it) 17. La France (Hybrid Tea 1867) 18. The Doctor (Tea Hybrid, 1936) 19. Marchesa Bocella (Hybrid Perpetual, 1842) 20. Barbara's Pasture Rose (Hybrid Perpetual, found rose) (Thank you Rosefolly) 21. Enfant de France (Hybrid Perpetual, 1860) 22. Duke of Edinburgh (Hybrid Perpetual, 1860 to 1869) 23. La Reine (Hybrid Perpetual, 1842)

I would remove a lot of the old gray wood next March--cut the canes at the ground or at the graft if one is visible. Hopefully it will respond by making some strong shoots from the ground. It looks like this rose is best grown as a climber with all the strong shoots fastened to the fence. Tie them rather than weaving through the lattice. Create a fan pattern by leading the first canes low and almost horizontal, the next in a low arc above the first, etc. Having lots of cane nearly horizontal (or 45 degrees) will greatly increase the amount of bloom. Most roses have canes that are best trained after they have grown well out and become fibrous. Then they will bend without breaking. After it begins to get crowded again, remove a fraction of the oldest canes each year at the base, Do the main pruning in June after blooming to maximize the amount of bloom. You can also remove the stronger laterals (strong secondary shoots off the main canes) unless you need them to fill in. Cut the shorter laterals back to 2-3 leaves after blooming. The spots visible in pics 6 and 7 are blackspot disease, but the plant must be somewhat resistant or it would be defoliated by now. The foliage appears to be of modern type. The hooked thorns suggest the parentage includes climbing species. You can show us blooms and buds next May.

Sylvia while I can't help you identify your rose other than a wild guess of Mr. Lincoln. I would suggest contacting your local American Rose Society they have consulting rosarians that would be happy to come out and look at your rose. They could also suggest how best to prune and fertilize it. the other alternative is you wanna take a, wack at it yourself, is to cut out the three D's ,as Baldo likes to say, remove; dead, diseased and damaged canes. Damaged canes are often damaged because they are rubbing against another cane so removing one will help open up the plant, and in doing so will likely improve and invigorate the rose. Those are the basics but practically every rose book out there will further explain how to properly prune a rose bush there are also a number of helpful you tube videos specifically on pruning.