Also should have mentioned to make sure that the Project tool window is set to Dockable, that would also prevent it from docking into the tab-group. You should be able to check that from from context menu in the Projects tool window. If you are unable to get it to come up, for whatever reason, you can go the main menu, View > Toolbars > Customize..., and check the Allow Docking checkbox for Projects.
The last thing to try, if all else just isn't working, is to go to the command-line (Esc key in most emulations, or just mouse click in the message bar area at the bottom), and type: tbResetAll. This command resets all the toolbars and tool windows to the factory installed default.