The second half of Puzzle mode involves clearing the same figure twice. The figure is considered "cleared" as soon as the timeline passes over the END of it. However, only blocks the timeline passes AFTER you've completed the figure will actually be deleted. If you wait until the timeline has passed over most of the figure before actually placing the final block, you will accomplish the first clear--but only the rightmost edge of the figure will be deleted. For most x2 Puzzles, this makes it ridiculously simple to complete the figure a second time and clear the stage.
There are 39 unlockable skins in this game. You have to play through almost all modes and master most of the game before being able to get the last ones. In the US version of the game you get random set of skins through certain level ranges instead of set levels as described here. But the sets are not completly randomized and the later skins are still obtained in the end.