What is it's intended purpose? Do you want it in good enough shape to use or just display? The trouble with cutting a section off is that you will have to secure the ends of the quilting threads so they don't unravel. If you do that then you could use the damaged part to repair the rest of the quilt. I would only use the extra fabric to applique over the damaged spots rather than trying to piece them in. You want to be able to save as much of the original quilting as possible.
If it's just for display, I would gently submerge it in a bathtub with mild soap and gently manipulate it and rinse it in the tub too. I'd let the quilt drain in the tub undernight so it would not be soaking wet before you try to lift it out. See how it looks after washing and lay it out flat on a sheet to dry. It's not a really practical time of year to attempt this if you live where it is cold and it might be better to wait until spring/summer when you can lay it out outside. If it's just for display I'd fold it so the "good" part was out and not bother to repair it but do what you think is best.