Not really. Usually the phrase hon har gått bort, is used which basically translates to she has gone away or she has passed away. It is considered the most polite way to address the death of a person.
Swedes in general don't tiptoe around the subject of death that much though. People can be pretty blunt even when talking about very serious issues.
I also like the slang expression hon har trillat av pinn, literally she has fallen off the branch.
No, if she rests "i frid" she's dead alright, we just don't use this expression very often to say that somebody died.
I have a swede friend and when I told him that my granny died, we fight because , he was so cooooold about it. Sometimes I dont know when his behavior is cultural or personal. Now you claryfied this to me. Extremely different from my country. People would cry with me and make everything I wish, trying to confort me.
Definitely a cultural thing. If someone tells me someone close to them has passed away I acknowledge it by saying Jag beklagar or Jag beklagar sorgen which is equivalent to the English I am sorry for your loss.
Other than that nothing really happens. My grandmother would quite often joke about her coming death and even that was not a big deal to us.
I should add that while common, it's improper etiquette to beklaga sorgen - you're expressing sympathy for the reason somebody is being sad, not for the sadness itself.
Expressing sympathy for the loss works: jag beklagar din förlust.
I personally use med varmaste kondoleanser = with warmest condolences.
This page is pretty good, though it is, of course, in Swedish: https://fenixbegravning.se/hur-beklagar-man-sorgen/
Also in italian is "RIP" (Riposa In Pace), from the latin "RIP" (Requiescat In Pacem).
I don't know why somebody downvoted you. If you used this in Swedish for somebody not being dead, people would think you're daft.
I would guess it is because they replied in earnest to someone who was clearly joking, rather than due to the correctness or otherwise of their statement. ;)