Can the iCloud share app-specific data across devices? Like many people who own both an iPhone and iPad, I like to play games on them -- preferably on the pad when I have it at hand, but on my phone when not.

One thing that always irked me was that, for games that run on both, progress on one device wasn't saved or shared to the other.

If I configure both devices to use iCloud, will they synch game data? (And if so, can I specify which version of the data takes precedence?)
The direct answer to your question is It depends.

Certain applications like iWork can store data in the cloud which is then synchronized across devices. However there is currently no interface to allow you to choice which document is the primary, as the data is synchronize on a last updated cycle.

If the game support using shared document via iCloud, you would be able to synchronize game data, however, by default it will not.