I want to program an application that changes spaces the way DVD Player does on OS X (screenshot of the transition below, but it may not be helpful because I cannot get my timing right). How can I make my application open a window, and move into a different space the way DVD Player does?

That’s a new feature in OS X Lion. It’s called Fullscreen apps. You’ll want to look at the documentation for how to get your app ready for Fullscreen:
http://developer.apple.com/library/mac/#documentation/General/Conceptual/MOSXAppProgrammingGuide/FullScreenApp/FullScreenApp.html