I'll second the clean new sleeve argument. With sleeves with art or other printing on them, I'll put the disc in the new sleeve next to the old sleeve in the jacket. With old, plain white sleeves, I just toss them.
On the second mat, I do the same thing - lay the backed mat on the regular, "stuck on" mat so the clean side doesn't touch the dirty mat. You have fewer threads to grab with the clamp, but it should still work.
For rinsing, I do another round of scrubbing, more lightly, with distilled water and a dedicated rinse brush. I feel that a squirt bottle doesn't get into the grooves enough. No evidence to back that up.