Sure there are! You just have to find the ones you like, just as in any other stories.
I think that harrypotterwiki.com is pretty good.
I don't have the ability to read fanfiction, but there are many great Spuffy AU fanfiction stories available online. Websites like Archive of Our Own or Fanfiction.net are good places to find new and interesting stories to read.
"FanFiction Royalty" is a term that refers to popular authors within the fanfiction community who have gained significant recognition and influence due to their large fanbase, high-quality writing, and original storytelling in the fanfiction world. Examples may include authors such as Cassandra Clare, Rainbow Rowell, and E.L. James.
As in Doctor Who stories? Fanfiction is a good website for any/most TV Shows, books, etc, Doctor Who has tens of thousands of stories on Fanfiction so check it out!
You might be able to find it on any number of fanfiction websites: Wattpad Fanfiction Archive Of Our Own Quotive Tumblr
Very, very rarely will fanfiction turn into an official movie screenplay or television teleplay/script. The only way it could potentially happen (on purpose) is if the writer in question is also a screenwriter for the film as a franchise (you can't exactly write fanfiction for a movie that's already come out, and then expect the fanfiction to somehow work its way into a finished movie--it would have to be for a subsequent movie) or the television show.However, most of the time, writers are forbidden by their legal contracts from reading or publishing fanfiction for the fandom that they are directly responsible for creating. This doesn't mean that writers don't know about fanfiction, just that legally speaking, they can't issue any official statement on it (such as "This is such a good fanfic, it should totally be one of our episodes!").
You can upload your Vampire Knight fanfiction on websites like Archive of Our Own, FanFiction.net, or Wattpad. These platforms allow writers to share their fan works with others who are interested in the fandom.
This term is usually seen with fanfiction stories and a non-canon in reference to Harry and Draco means that this pairing was not promoted in the original books, but was created by fanfiction writers.
Fanfiction.net is always a good place to start if you ever want a fanfiction.
Some helpful fanfiction sites are Fanfiction.net and Archive of Our Own. Links to these can be found in the related links.
Fanfiction is fiction written about someone else's universe, so the main character names are already going to be there. You just have to pick something to write fanfiction about! If you're looking for good names for your original characters, here's a link to a good name generating website.
There can be no generalisation about mathematicians or any other professions. Even some authors are not good writers!