What parts of the universe are shared between characters?
The Universe (Planets) carries over between characters. Every planet and every structure on it is the same between all your characters, even if you delete your old ones and start all-new.
That is because the Universe is saved in a seperate file that is independent of your characters.
That means you can either reset the world for all your characters or not at all. That might sound bad, but considering there are millions of planets in the game, you shouldn't worry about running out of stuff to do.
EDIT: Ships do not carry over as they are saved with your character