What should I use for a sandbox bottom?
Since you are going to place it on a deck, move it around, want to keep water and cats out, I would just buy a plastic one with a lid.  In my opinion its not worth the hassle.  The wood alone will make it heavy and hard to move around, let alone with sand in it.  

They are fairly cheap, we bought one a few weeks ago at a garage sale for $5.  I know there is pride in doing it yourself, I wanted to build one too, but in the end, it may not be worth it.