"My grandmother died the day before yesterday."
I don't think it is an appropriate sentence to use. Nobody use 死ぬ to talk about the death of people they know. It is so direct that it is not respectful. Check with a native before using it. (I am not native.)
亡くなる, 逝去する are more appropriate.
For family members it is not so unusual to use 死ぬ. As you mentioned, however, it does sound a bit too direct. For this reason many people have stopped using it (even for family members) in favor of 亡くなる.
Like BJCUAI said, I think it's the same as with English. In that English sentence, the word "died" isn't really appropriate either.
My grandma passed away recently (funeral is actually today) and I sometimes used the word "died" when talking to people with whom I speak very casually, but with other people towards whom I think more about the words I'm saying I concentrated on saying "passed away" (especially when I spoke to people who are elderly or who know people in ill health themselves).
I believe it's the same situation with saying 死ぬ or 亡くなる. If the purpose of this sentence is to teach the word 死ぬ, I think it would certainly be better for it to be changed to a pet dying rather than a grandparent, but I don't think it's unused the way it is, just a bit blunt.
I agree. This is odd. Seems like new since the update? I think previously they did say なくなりました？
Previously, Duolingo used 亡くなる in the sentence きのうそふが亡くなりました。So it really boils down to whether or not Duolingo wants to be consistent with itself. Grammatically, both are correct.
It is おととい. The links below are to two E-J/J-E online dictionaries' entries for 一昨日. They both show furigana. You should bookmark one of them (or find one that works for you) so you can find out readings yourself right away.
That's not how it's pronounced in the audio for this sentence. Is the audio wrong or is that a different reading?
EDIT: I see that the first link also gives いっさくじつ as a pronunciation, which does match the audio.
いっさくじつ is not incorrect, but it is extremely stiff language and unusual in the extreme to hear in everyday conversation. Usually it is reserved for news broadcasts, formal presentations, etc.
Given the context (a family member dying), it is not unnatural to use this form, but I would not expect an entry-level program like DL to introduce that reading. Most people use おととい or おとつい (regional).
Ok, to suddenly give me a bunch of kanji I have never seen before and expect me to know them without cheating is not fair.