Wii Virtual Console — Black screen

No, there are no games that require a classic controller but do not work with either a gamecube controller or the wiimote & nunchuck combination.

There are games however that only support two out of the three combinations, most notably, Monster Hunter Tri, which only supports the classic controller and the wiimote+nunchuck combination, but do not support the Gamecube controller

Do note however that some games are not optimized for the gamecube controller and play a lot easier on the classic controller (SNES games notably) and the gamecube controller is the only controller that allows you to play gamecube games.

I wasn't able to find any way to transfer the data from my old Wii to the new Wii U without running the Wii U Transfer Tool. I tried copying all my Miis to a Wii Remote but the new Mii editor under the Wii System Menu on the new Wii U doesn't offer a way to transfer to/from Wii Remotes. As expected, the Wii U Transfer Tool deleted all data off the old Wii.

However, I found a kludgy workaround: Before using the Wii U Transfer Tool, I had copied all my Miis to a Wii Remote, and I copied all my save game data to a different SD card first. It was a major pain, but it did allow me to keep a backup copy and restore the Miis to the old Wii after using the Wii U Transfer Tool.