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."

I think it's because 'дедушка' is masculine, even though it ends with -a.

Because it's in accusative case. It's a masculine word but has a feminine ending, which means that it's declined like a feminine word, but determiners and adjectives attached to it decline according to masculine tables.
Masculine - Папа: Анна спросила этого папу - Anna asked this dad
Feminine - Мама: Анна спросила эту маму - Anna asked this mom

This set-up makes sense - there's no "-a" masculine or neuter nominative ending, and the only available declension rules are those for feminine words ending in "-a". But since it is masculine, words which attach to and thus have to agree with the masculine noun are declined according to masculine tables.
Examples:
эта молодая мама - this young mom
этот молодой папа - this young dad

Could it also be "my grandfather" and not Anna's?