Yes you can play gamecube games with the classic controller. The classic controller was practically made to play old gamecube games along with the arcade style games found in the nintendo shop.
No, the classic controller is made to emulate the feel of an older Nintendo controller while playing Wii games; however gamecube games can be played by plugging in gamecube controllers.
Yes, you can either hook up your gamecube controller to your wii, or use a wii classic controller to play gamecube games on your wii, but you cannot hook up your wii classic controller to your gamecube.
