If I were to play a Wii game on the Wii U, could I play it on the tablet? Using the tablet as the display and use the normal Wii controller+nunchuck to play the game? Or does that not work for some reason?
Unfortunately, the Wii U screen is a little more complicated than simply streaming from the main unit, and thus you will only have the ability to play from it on games where the developer has taken the time to implement that.
Since Wii games were (understandably) developed before the Wii U came out, none of them have the code necessary to transfer to the tablet. It remains to be seen if Nintendo will "retrofit" some (if any) of the virtual console games to use the tablet.