The sticker museum acts as a sort of reference for "Things" and "Battle" stickers that you come across in the game. The description of "Things" tells what area the sticker was found in. The description of "Battle" stickers offers a power rating and details what sort of enemies it damages. The description for all sticker displays offers some sort of flavor text.
For completing the sticker museum, you are rewarded in two parts:
- All Things Stickers will unlock a "sound test" room in the museum where you can listen to the game's tracks. It is set up like a playlist, which you just parse through until you find what you want to listen to.
- All Battle Stickers will unlock a room where you can view enemies for 1 coin.
When everything is said and done, you have a short conversation with the curator toad and he will respond to you differently in the future.
Speaking to your particular concern: The museum seems to be there largely for re-playability and completionists, so there isn't any rush to get your more valuable battle stickers there. If I find rare stickers in a level, I usually go and stick them there when I'm done and then return to that level to retrieve another copy of that sticker. Since they respawn in most circumstances, there isn't any major risk in doing that.