Nope, you can't see the code from account management, and I doubt ANet will let you know it either.
Thenagain, you don't need to know the CD key once you have used it, it's tied to your account, and that's that, you'll never need to input the key again, even when reinstalling.
Thee ONLY reason you may need that little piece of paper is if someone steals your account and ArenaNet asks for proof that you own that specific account.
Use a secure password and DON'T use the same password that you use for your e-mail account.
I wouldn't worry about the slip of paper. You should never need it. Even if someone steals your account, send arenanet an email and they can use your email address to confirm your account, even if it's been changed by the account stealer.