I haven't done any metal spinning since high school, about 50 years ago. But I think the proper term for metal spinning is mandrel, but die works for me.
High temperatures? If things get really hot then you are doing something wrong In the annealed state, most tools steels turn pretty nice, but I think 1018 would work fine for a mandrel, many times wood is used for the mandrels. Going from a tube to a cone should be pretty easy, you are really not moving that much material.
You might do a YouTube search for metal spinning. Some pretty good videos to give you some ideas.
Here's one: