The name of the star that Earth orbits is "the Sun". In particular, its name is not "Sol".
Don't take my word for it, of course. Just look it up in the dictionary.
That's fiction. If the Sun is called "Sol" in a work of fiction, that just means that's its name within the setting of the story. In the novel 1984, England is called "Airstrip One", but this in no way means that this is really England's name.
Fiction authors are generally well aware of this distinction. The Sun is called "Sol" in Isaac Asimov's novel Foundation, but when not writing fiction, Asimov always referred to it as "the Sun". See, for instance, his textbook The Sun.
No, that doesn't apply to proper names. Specific people, places, and things do have official names, which can't be changed by common consensus. If we all started calling you "Orville", would that mean that your name is "Orville"? Of course not.
Also, even if that were the case, nobody actually calls it "Sol" outside of fiction, except science fiction fans in informal settings. In formal nonfiction publications that have undergone editorial review, it's called "the Sun". Some examples:
No they do not: example 1, example 2, example 3.
The International Astronomical Union, an internationally recognized group of professional astronomers. And they call it "the Sun" exclusively (example).
True but that doesn't matter, since I'm talking about the English name for the Sun. Other languages have other names. The Portuguese word for Norway is "Noruega", but that doesn't mean the actual name of Norway is "Noruega" when you're speaking English.
True but that doesn't matter. "Natal" means related to birth, but the word is still "birth", not "nat".
Fine. I don't care. Just don't tell anyone else that its name isn't "the Sun".