The first Quilt (the blue one) was not wide enough. When on my side, the edges were up off the ground. So, I added a "floor" as you mentioned. But, it was not necessary, so I cut it just to leave the flaps. The flaps are just one layer of tapered nylon. The black quilt was originally flat all the way, but I since added the footbox to make it warmer.
In short, I would add the draft stoppers to the side. Slipping off the pad has not been a problem. I tried velcro straps on the first one, but found them not necessary.