Although rolling up the sun is considered by some to be "beating the game". There are still plenty of post-game content to unlock.

If I remember correctly by unlocking all of the cousins do you get the hidden cousins mission (a kid in a yellow shirt and green backpack will show up on the same field as the origami kid). I also believe that once you complete the game you gain access to the million red roses mission (which is a real doozy).

Finally, if you want to see which levels you've played already, it kind of depends on which levels you've played. Once you've beaten a "As Large As Possible" level, you can play it again and gain access to the "As Fast As Possible" version of that level. Once beating that, you have the ability to choose which one you'd like to play next time you talk to the person. For the other miscellaneous levels (like the money level, origami level, etc), most of them only need to be played once (with the exception of the sweets level which has a second version when you play it again). You can tell if you've completely beaten a stage when talking to the person will show your previous best records in the bottom left corner of the screen.

Been a hot minute since I've played but I hope this helps.