Some popular ones might include those where the author reimagines Harry in different historical settings. For example, a story where he's a knight in medieval times. Another could be a modern - day AU (alternate universe) where he's not a singer but a college student dealing with normal college life problems while still having his charming personality. And there are often those that explore his relationships with other characters in a really in - depth and emotional way.
Twilight has nothing to do with Harry Styles fanfics. It was created by Stephenie Meyer as an independent novel series with its own unique story and characters.
It could be trending because it features a new take on Harry Styles' personality or it offers a fantasy storyline that fans haven't seen before. Also, good promotion and word-of-mouth can contribute to its popularity.
I don't know for sure. It could be free or have some kind of cost associated. You might want to do a search on popular content platforms to get a clear answer.