Q: Icloud and facetime popups

Have you tried entering the password?

The same thing happened to me, when I put in my password, my icloud account was hacked into and several emails went to people in my address book to sell diet pills.....I changed my password, and called apple.  They had me do several things in the settings, and I'm still getting the popups.

I had this problem on an iPad.  After many (many) hours of research and frustration, I managed to contact Apple Support chat who helped me to solve the problem.  Hopefully, a couple of key lessons from my experience will save you many, many hours.

In my case, the log in messages were appearing because I was still using IOS 7.X on the problem device.  Once I upgraded to the current version of IOS, the problem went away (after a bit of extra mucking around).

You need to do several of the following tasks on a PC / Mac, as the problem device is unusable with the incessant messages:

1. If you are using the app Find my iPhone on the problem device, you need to log on to icloud.com on your PC/Mac and remove the problem device from Find my iPhone.  Do this BEFORE proceeding to the next step.
2. Connect the problem device to your PC / Mac (preferably the machine that you synch / backup to).  In iTunes on the PC / Mac, go to the screen showing your device, go to the Summary screen and select Update.  This took some time, and at one stage iTunes on my PC asked me to enter a password on the problem device, which I couldn't because of the incessant iCloud log in messages.  Somehow, the update proceeded anyway.
3. Once the update is complete, you need to step through a few screens on the iPad (pretty straight forward - similar to screens seen when first setting up the iPad)
4. If you get to the screen to sign on to iCloud, you can try.  If it fails (ie the terrible messages start to appear again), Skip this step.  Note it may fail because you are no longer connected to WiFi (see step 6)
5. Continue to step through the setup screens until you arrive back at your normal Home screen.
6. Before doing anything else, make sure that you are connected to WiFi / Cellular data on the problem device.  This may have disconnected during the update process.
7. Once you are connected to WiFi / Cellular data, you may have to enter your iCloud password, but it should work this time.
8. If you are still getting the log in messages, go to Settings, iCloud.  Scroll to the bottom of the screen and push the "Sign Out" button.  If the option appears to keep your data on your device or delete it, select Keep data on device.  You may have to enter your password a few times here (but have faith, you are nearing the end of those infuriating messages)
9. Restart the problem device by holding down the power button for about 10 seconds (until the swipe confirmation appears on the screen), then after the device has completely shut down, restart it by holding down the power button for a few seconds (the apple logo should appear on the screen).
10. I hope these 10 steps delivers a device to you with NO MORE INFURIATING LOG IN SCREENS!

A couple of other learning points.  Contacting Apple Support can be a challenge, as you either need to connect to a link sent to the email address on the device, or you need to enter the Serial Number of the device into the Apple Support page.  You can do neither when you are getting the incessant log in screens on the problem device.  You can get the Serial Number from iTunes on the PC / Mac that you use for synching / backups.  It can be found when the problem device is plugged in to the PC / Mac, in iTunes on the device Summary page.

That usually happens when your account has been disabled/locked because of security reasons i.e.. someone tried to login to the account multiple times with the wrong password.

Go to:

http://iforgot.apple.com/

and unlock or reset your password.