Firstly welcome to the forums.
1. Yes the Nunchuck attachment is wired and attaches to the Wii remote, but it does not plug into the Wii console and communicates wirelessly.
2. Some games do require the nunchuck attachment (Red Steel, Zelda etc), but for example Tennis and Bowling (WiiSports) only require the Wii remote itself.
3. Not too sure what you mean by traditional? Currently the Wii Remote and Nunchuck are the only controllers used for Wii games. Are you referring to the classic controller?
If so that controller is designed purely for use on the Virtual Console.
Hope that helps.