For all 16 main stages, you can unlock Eternal Mode (limitless play) by achieving at least an 80 as your second or first score on each level.
Clearing each stage will unlock that stage's song for use in the Sound Test. By clearing the entier game, you can get the songs Blue Orb and Song for King of Kings.
In order to access the bonus 2D Katamari stage, simply complete the final Crown stage. The King of All Cosmos will speak and everything will suddenly turn 8-bit. Play through the stage as the credits roll.