Is it possible to customize application icon badge: Xcode?

Not possible at all. You have to use Default. After the app launch you can show custom badge icon of the UITabbar.

All the best !!!

If you mean the badge number that can be shown on an app icon, in white text over a red circle, then no, it cannot be set to a custom image. You can only control the number that is shown. It is set using the -[UIApplication setApplicationIconBadgeNumber:] method, i.e., the applicationIconBadgeNumber property on UIApplication:

[[UIApplication sharedApplication] setApplicationIconBadgeNumber:99];