If they are good friends then yes she will be in mourning because she most likely knew her (there's also a bug going around that Sims will grieve a sim they barely know.) but a good friend means high relationship so yes she will mourn. They said grief will act like rl so it will take time to heal.
I had a new Sim that missed out on the 'Welcome Wagon' and is now mourning someone I don't think they ever met - and I mean mourning them hard. Suffering sadness, getting various 'grief' related moodlets almost every day. Finally got so irritated I deleted them.
That's the bug they not suppose to mourn a person they barely know report that bug you're probably not the only one.
Sure but the dead sim is technically still alive because she has to show up at her job still. I guarantee you when I die I will NOT keep coming to work.
She's a ghost yes but she died so the grief is still there just like my Sims father he died he's a ghost but the Sim still died that's where the grief takes place.
