Any suggestions for some challenging climbs anywhere in South America during that time?
Try the mountains around Salta or Arequipa. They should be in season then and should be compatible with your experience.
Any ideas on a particular trek or region?
The Atacama and around Salta is usually in season then.
Ecuador can be reasonable, though it's not the ideal time to go.
Most other areas, October-November isn't the ideal time to go. Central and South Chile/Argentina might still have some good snow routes.
Basically, these are the ideal seasons for each part of the Andes (off the top of my head so I may have missed some):
Venezuela, Colombia: Mid-December through February (or sometimes into March).
Ecuador: December, January, June, July, and August.
Peru (non desert areas): Late May through August or September (June and July are prime).
Peru (desert areas): March through November.
Bolivia: Late May through August or September (June and July are prime). March through December are still good for the desert areas (such as around Licancabur).
Atacama Desert; Chile/Argentina: October through February, but this varies by the height of the peak (because of cold).
Salta area (Argentina): October through December.
Central Chile/Argentina through southern Argentina (unless skiing): December through early March.
Which areas would be good to get in some rock climbing on a regular basis?
Patagonia, Bariloche, and the Tres Cruces (Bolivia) are all good areas for this. Parts of the Cordillera Blanca as well.