IPhone/iCloud help!

Probably not. Gamecenter data is completely separate from game data, so it has no bearing on this issue. Other than that, I'm an android user so I can't say for certain, but don't you have to activate iCloud on an app-by-app basis?

I don't think so

You've got to activate "Documents & Data" in Settings > iCloud to activate syncronization AND the app must use this system feature.

IF you have an iTunes backup with the game state in it, you can apply it to your iPhone (if you think that's really worth it) or extract certain data (e.g. with iPhone Backup Explorer) from it and then use iFunBox to put the files into the right directory.

I don't see any other solution (And yes, gamecenter is kinda stupid, ...)

If theres no button for it, nope.

The only way to save your actual game data is if the app offers an additional service to save your data through the cloud. Otherwise only achievements will be saved. Angry Birds, Major Mayhem, etc. are examples of games that save your data to their own servers. You can also back up save game data to your computer through iTunes. Hope this helps.

EDIT: There may be some iOS games that have the ability to save to iCloud/Game Center without turning on Docs & Data (which is default off because it takes up iCloud storage wicked fast), but I haven't seen one in a long time. Now that I think of it, Major Mayhem might use this utility. What I said before still applies too.