"My grandmother died the day before yesterday."
Like BJCUAl 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.
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.
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).
If you mean 'shinimashita', it is acceptable for both humans and animals. It is just more direct and blunt. If you want to be more sensitive and indirect you will use 'naku narimashita'.
The animal vs. human example is just to emphasize that we don't normally use the same caution or sensitivity when mentioning the death of animals as we do with humans.
You are talking to a third-party, likely a non-intimate, about your own grandmother: Use 祖母.
Talking directly to your departed grandmother (at the tomb, etc.): Use お祖母ちゃん.
Talking to someone intimate enough to consider your grandmother like their own grandmother (or vice-versa): お祖母さん・お祖母ちゃん。
Talking about someone else's grandmother in general: お祖母さん.
These 'standards' pretty much apply to distinct ways of saying brother, sister, mother, father, etc.