What category would a quiz about Apple Inc be placed?

That would depend on your exact orientation, but you are discussing the corporation as a business, it would go in World, where there are quizzes on Business World - Corporations; no entry yet for Apple. If you actually want to focus on the products, however, it might fit better in Science and Technology - Technology and Computers (and depending again on your exact focus, there are a couple of possible spots there - I see Macintosh, General Computing and other options in Computers; if you're looking at a range of products, maybe General Technology is the place.

I have a category question, too. Where would a quiz with quotes by sports figures—not just athletes, but also announcers, commentators, etc.—belong? Should it go in sports, or should it go it Quotes (under People)?

Although I do not edit in either category, so cannot give definitive advice, I would suggest that it may depend on the nature of the quotes. If they are sports-related, then it definitely goes in Sport (as the People category states, "Quizzes on sports figures are located in the Sports category"). However, if they are more widely philosophical in nature, and the sports aspect is just providing a unifying theme, it might fit in Quotes.

That'll be fine in sports (it's not really a People Quotes quiz).