I'm running 12.10 and in 12.04 I was able to move the items (into other categories than where they're presently situated) in the applications menu by means of moving them in the application, main menu.
--- this problem was solved in the chat ---
The problem was that the file /home/brentonhorne/.config/menus/applications.menu wasn't writable by the standard user. brentonhorne is the standard user on this system, for everyone reading this, you'll have to exchange this username of course.
The problem was fixed by setting the permissions of the .config folder and all of it's subfolders as they should be (owned by the user, readable and writable by the owner) - again for everyone else, you'll have to insert your own username for brentonhorne.