Apple Account Verification
First, check and see if you can log in to your account at appleid.apple.com. Support can help you a lot more if you can log in here, since you can get a support PIN (which acts as a password). 
Tell them your support PIN (found at the very bottom of the page) and they will know that it's you and can fix your account with only one security question.



Your Apple ID should also be set up with a non-apple email address, and that should be where you are getting your codes. Check all of you emails, spam folders and all, to make sure you didn't miss an account reset email. Good luck!