I've done a number of these types of projects and it is always a bunch of "best/worst case" decisions. Typically I will leave on the old back, just putting on a new one and new binding. I typically will do minimal machine quilting to hold the layers together, griding the blocks or just a simple pattern leaving the old hand stitching in place, if it was tied originally I usually take out the ties after I stitch it down but sometimes there are large wear/tears at the ties. In your case, it does sound like maybe a new thin batt may be called for.
I usually don't have any problems with the fabric shredding, but it all depends on how bad it is. I usually use a denim weight needle for machine quilting on my regular machine, but maybe a lighter needle is called for. You may want to put a layer of netting/a fine thin fabric you can see through over the top. When I've had to replace fabric, I will try and get as close as I can with my vintage fabrics but I typically distress the fabric a bit by over bleaching it first, while the back is off you can also do some extra stitching through the most worn areas, either zig zag or just inserting a similar colored fabric (maybe a light batiste weight solid) underneath the original fabric that you can stitch through.
Good luck. Remember that it is the memories that are most important, our workmanship should be the best we can do but no matter what we do chances are it will be noticeable. Some times the quilt is so worn out that really the only thing that can be done is use the whole thing as batting in a new top or store it as is and only take it out when we need the hugs the worn fabric can give.