To play as a different character (not the green dragon) in a puzzle game, press the following when you are in the starting circle on the map: Leftx2, Up, Down, then L1+L2+R1+R2 together. A character select creen should appear and you can then choose other characters.
At the beginning of the game (when the "Press Start" thing is flashing), press R1, Up, L2, Down. You will then see a small green guy that will appear in the lower right corner. Then when you start a puzzle game, there will be new Backgrounds. @_@