Also asked, does iCloud backup my game data?
The only time you can restore apps from an iCloud Backup is when you're setting up an iPhone or iPad. Next, you just wait until your iCloud backup is restored. All of your apps will download from the App Store automatically, and the backup will restore all of your app and game data.
Furthermore, how do I save game data on my iPhone? Backing up your iOS game dataSelect the device and open the Apps list. Select the app you'd like to back-up data from. You'll see a bunch of folders, including Documents and Library. Somewhere within Documents and Library will be the highscore data, and the location and filename varies by game.
Keeping this in view, can Game Center save your progress?
Game Center does not currently have any mechanism for saving game progress. For games which store progress information on your device, that information will be deleted when you delete the app. However, it will be backed up in iTunes, so you can restore this from a backup (see this question for more information).
Is Game Center connected to iCloud?
Before iOS 10, Game Center was Apple's gaming-themed social network that connected through your iCloud account: It was built around a standalone app that let you add friends, challenge their high scores, and invite them to play games.