I'm not sure what kind of damage y'all are picking up on your (full) mats. I have been storing them vertically under my mat table for years with no ill effects.
I only wish that some of the distributor employees that fill my orders would take the same precautions when handling suedes and all the other mats I buy from them. Bent corners, scuffs from dragging another matboard across the colored paper surface, knicks in the edges from dropping them down on the edge of a bin or table or loading dock, stuff like that. Sloppy handling has lost more than one distributor my mat business. There are too many other distributors that want my mat business and are willing to keep some kind of quality control over how their people handle the boards.
Regarding cutoffs of suede boards, I sort them in reverse order by size. In other words I put the largest pieces to the left in the bin with the suede facing left and smaller pieces graduated to the right on down to the smallest pieces. That way there are no edges leaning against the face of the adjacent board leaving a line or flat spot in the suede nap.
If you do get a mark in your suede board, try using a stiff bristle knuckle brush to massage them out, the kind of brush you use to clean your nail area and knuckles. I found that using a stiff bristle brush such as described in a circular action will take out most all marks and flat spots and renew the mottled look of the suede.
Framerguy