I've never had someone close to me die, so I don't know how I would react. I think I would definitely be very sad, but again, only for someone close to me.
I don't care about the deaths of people I don't know, though I do recognize that if a friend's mother dies it will make them upset, and so then I'll be upset because my friend is acting upset.
When my dog died a couple years ago I wasn't sad because it wasn't a surprise. He was a very old dog who was slowly wasting away for quite a long time, and I was actually relieved when he died, because he wasn't in pain, and I didn't have to change his diaper anymore. It was his time to die.
I do remember that my mother came to wake me up when he died, and cried for a while. She was more attached to him than I was. Even though my mother doesn't think she's on the spectrum, I'm convinced she is, and I think it was very aspie of her to check the clock as the dog died and call out, "Time of death, 8:36 AM." (She's obsessed with medical TV shows hahaha.)