Melissa and Doug has puzzle shelves. I use one of them and I also keep other puzzles in a basket and put the pieces to each puzzle in their own zip lock bag.
I bought some on ebay a few years ago. I tried searching but can't find them I think they were "media bags". The library uses them and I thought it was genius.
Re: Puzzle storage
Do you sew? You can make your own cloth bags for them.
What about just using kid sized pillow cases?
We use something like this, but ours is just a plain wood color.