Not only can you use it, but I actually recommend it for the best game experience. I actually prefer it to the Classic Controller. I'd say get a cordless one if possible though, because the cords are never long enough and the cord extensions are garbage.
yes, you can. I know because whenever I play I use the gamecube controller. And, you can play with up to 4 people using both types of controllers, for instance, if you have 2 gamecube controllers and 2 wii remotes, all 4 of you can play together.