"Install Xcode" in Applications, necessary after I've installed Xcode? Do I need the Install Xcode after Xcode is installed?

Do you know of any reason it installed an installer to Applications?
You may want to keep it in order to be able to download delta updates instead of the full version when updating via the Mac App Store.

If you don't care about that or care more about disk space, you can delete it (you'll be asked for the administrator password) and continue to use Xcode.