If I delete my Facebook account, what happens to my Dragon City account? Is there any way I can delete my Facebook without losing my account?

If you delete your facebook account then you need to link your game to another of the other options. You can link it to your game center account on ios, your google account on android and your windows store account on windows 10. If it is only linked to facebook then will lose access to your account if you lose access to the device it is installed on. For example if the device is lost, stolen or broken.

Also keep in mind that the game can only be linked to one facebook account, so you will not be able to link it to another facebook account.