Your Facebook public user name will be username of your Facebook email, by default any can one can email to you at email@example.com, If you want to control who can send you Facebook messages and email here is how.
Turning off Facebook email
1.Sign into your Facebook account, click Account drop down menu select “Privacy Settings”.
2.Select “view settings “ in the “Basic Directory information” section.
3. For “send me messages” setting choose “Friends only”. Or you can choose “Friends of Friends “ if you wish.
If you choose ‘Friends only “ Facebook won’t send messages from people other than your friends in Facebook, those senders will receive automatic bounce-back replies.
It can take 24 hours for selected messaging options to take effect.