I'd like to delete a game off of an iOS device but I would prefer to not lose my progress in case I want to return. Is there any way of checking whether the game/app puts the game save either in iCloud or another external service?
You can see a list of apps using iCloud storage from the Settings → iCloud → Storage → Manage Storage menu. But even if your app is listed, there's no way of checking exactly what data is saved.
If you back up the iDevice via iCloud, the game data will be saved there. You can also use iExplorer on a Mac or Windows machine.
This tutorial will show you how: http://www.pocket-lint.com/news/132888-how-to-transfer-app-and-game-data-between-ios-devices