Translation:Anna asked grandfather if he wanted to eat lunch with her.
How on earth does "не хочет" come together to mean "wanted"? Where did the negative (не) go?
Asking a question with a negative can be a slightly polite way of doing it, and it's done more often in Russian than in English. "Не хочешь пообедать со мной" is actually a pretty neutral way of asking "Do you want to have lunch with me?" If you use the negative in English, like, "Don't you want to have lunch with me?", it sounds as if you were expecting lunch, and you're surprised or disappointed. "Don't you want to? Why not? I thought you would."
Oh, I hadn't thought of "don't you want..." at all. That makes a lot more sense, thanks!
While I agree with you that it's less common, it would be appropriate in this sentence. If you're talking about a asking a third person to lunch, you can say "I asked him if he didn't want to have lunch some time".
So I think it should also be accepted in this case.