I think it is a programming error. The apostrophe appears in the answer sentence at the top of the page, but it is omitted in the word-choice exercise.
Yes. "At her parents' " with 'house' or 'place' implied but not written.
Listen more closely: Анна stresses the first consonant and она stresses the second. Duo has been consistent with these different pronunciations since the 'Alphabet' basics.
Boredom does tend to result from stunted observations. Or maybe you just need headphones... like Duo keeps suggesting.
No, you actually use the «у» preposition + genitive.
«У» has a number of uses. You must have seen it in the sentences like «у меня́ нет кни́ги» 'I don't have a book' (you could understand it as 'at my place, there's no book' if that helps).
when doing a listening exercise with this sentence, is there a good way to distinguish between "Анна" and "Она"?