i had the same iCloud-problem. I got crazy by getting the same Popup-question over and over again: log in to iCloud with the PassWord for non_existing_email_account@driving_me_crazy.com.
Finally I contacted AppleSupport and they gave me the solution:
- Go to https://appleid.apple.com/
- Click on: Reset Your Password
You get a new window with 'Enter your Apple ID to get started.'
- Fill in the emailadress non_existing_email_account@driving_me_crazy.com
> Next
- You get a new window with 'Please verify your birth date to continue'
> Next
- New window with: 'Select your authentication method.'
And choose for (this is the trick!!!):
Answer security questions: To access your information, you will need to answer the security question(s) provided when you originally created your Apple ID.
- Answer the two security questions.
> Next
Now you get the possibility to change your password for the emailadress non_existing_email_account@driving_me_crazy.com
After doing so I was finally able to login on my iCloud-settings on the iPad with the emailadress non_existing_email_account@driving_me_crazy.com. By doing so you get a prompt that your 'Find my iPhone' setting will be turned off and that was exactly what I needed all the time, as before all my iCloud settings (including 'Search my iPhone' were greyed out).
After this I was able to log out from the iCloud (scroll all the way down, you will find 'Log Out' written in red).
And finally, finally, I could Log In again, with my one_and_only Apple ID.
Good luck! Hope I didn't miss a step :-)