Passwords. The 'Facetime' and 'iMessage' passwords are the same; Your Apple ID password.
Apple ID is your email address. Is there any chance you registered your Apple ID using that gmail address?
See
https://support.apple.com/apple-id for more information and possible solutions.
Desk Icons. Try repairing your permissions using Disk Utility:
https://support.apple.com/en-us/HT201560
Have you considered upgrading to El Capitan? Improvement on Yosemite.