I unmounted all of my wood mounted stamps a few years ago. I took them to the bare rubber. I have strips of poster tape on my acrylic blocks to hold the bare rubber in place when I stamp. I can stamp a long time before the tape loses its stickiness and needs to be replaced. I have had no problems stamping a good image without the cushion backing.
When possible now I only purchase unmounted rubber stamps and most have the cusion backing on them. Since they don't need the poster tape to stick to the acrylic blocks, I use the other side of the block.
I used CD and DVD cases for storage when I first unmounted. I put strips of poster tape inside the case to hold the stamp. A couple of years ago I switched to binder boxes with page inserts like these (Stamp storage - Gina K. Designs, LLC
). They easily hold 4 pages of foam mounted or 5 of bare rubber stamps. The bare rubber ones still need a strip of poster tape. Depending on the size of the stamp set, you can often fit two sets on a page.
I did not have an index with images of each set so before unmounting a set, I put the stamps (block side down) on my scanner to get a picture of the set. For the ones I purchase unmounted, I scan the stickers or the case cover sheet. I print it on cheap cardstock to go in the case with the stamps since many no longer have a sticker and it makes it easy to see what's there.
Unmounting was a huge job that took a while but once it was done, I've been really happy I did.