Yes. They are all correct. In the U.S. no one would say, "I almost died of fear." People would say, "I was scared to death." (Even though they actually did not die.)

I almost died laughing. I almost died of fear. I almost died of hunger. I almost died of thirst. died laughing, died of hunger, died of thirst are all great. Died of fear sounds a bit odd to me, I'm not sure why! You could say "I almost died, I was so scared". 'I almost died of fear' almost sounds like something I would read in an old book. But see what other people's opinions are as well!

sentence two is ok but it is a little bit awkward, it does not quite have a true ring to it. " I almost died of fright" is the most common phrase we use but the other suggestions are also o.k. "you nearly/almost frightened me to death" = a common phrase. "I almost died" is a phrase you would say after explaining or telling a frightening scary story that happened to you.