I have on a number of occasions used painters tape instead of pins to keep the fabric laying flatter. Maybe that might work for you?
Maybe a sewing case or tackle box that you could keep close by you for your sewing notions that may hurt your kitty while you are sewing, that your kitty could not get in.
Also I wonder if eating small pieces of strings would hurt your cat, I know of many cats and dogs that eat long pieces of grass and the worst thing that happens is that they may vomit, but they eat the grass because they are not feeling well to begin with. Maybe ask a vet about this to see if it is a problem?
As for little fabric pieces I keep a lint brush handy to clean up little bits off of my mat and table, and my sewing desk when I trim my pieces.
I hope that you get enough solutions so that you can get back to quilting, I know that it is a sanity saver for a lot of us :-)