1. You can use the Wii Remote, Wii Classic Controller, or Gamecube controller. The last two work with all games, and the Wii Remote only works with certain games (it works with all TG16 games).
2. No, completely new ones.
3. No, but most are. Rayman for example, is not.
4. Yes, this is the prefered method of saving. You can copy/move saves onto an SD card, but I'm not sure if you can even load games directly from the SD card or not. Most saving is done directly to the Wii's built-in memory.
5. See above, but in order to play VC games, they must be moved to the internal memory. You can store games on SD cards, but they can't be loaded from there.
6. No, but it comes with 512 meg of internal memory (all you should need).
7. Yes, the Wii Remote requires 2 AA batteries, which are included with the system.