I hope I don't have to ask this in Sweden, I would not want to come across as incredibly rude.
It isn't always rude. What if he got into a car accident or something? Asking this question would imply that you are concerned/worried about him.
Yeah, pretty much. If you translate "is your grandfather alive?" directly, you would get "är din farfar vid liv?" but it's less common and a bit archaic.
I believe it also works for "is your grandfather living", which would be a bit closer to a standard English construction.
Seems to be a little bug: I answered "is your grandfather alive", and duolingo answered:"Another translation: is your grandfather alive"
I think there are two spaces between your and grandfather in DL's translation
What's wrong with: Is alive your grandfather? Or with: Is alive your father's father?
"Is alive your grandfather" has a word order problem. If you asked that, you'd be asking if alive, which is an adjective, is a grandfather. That is impossible.
It is the same problem with the second one.