Yes, there might be some out there. You can start by looking on fanfiction websites like Archive of Our Own or FanFiction.net. However, the quality can vary greatly. Some might blend the magical elements of Harry Potter with the supernatural themes of Twilight in really interesting ways, while others may not be as well - written.
As I mentioned before, Archive of Our Own and FanFiction.net are great places to start. These platforms have a vast collection of fanfics, and you can use their search functions to specifically look for 'Harry Potter x Jane Twilight' crossovers.
No, Twilight was not a fanfic of Harry Potter. They are two completely separate and distinct literary works with their own unique storylines and characters.
No. Twilight and Harry Potter are two completely different series. Twilight focuses on vampires, werewolves, and a human girl's relationships in a supernatural world, while Harry Potter is centered around magic, wizards, and a boy's fight against the dark wizard. There's no evidence that Twilight began as Harry Potter fanfic.
One common theme is the clash of different supernatural elements. Since Harry Potter has magic, wizards, etc., and Twilight has vampires and werewolves, seeing how these interact is a big theme.
In fanfics, the red string of fate can be a creative device. For Harry Potter, it could be a magical string that only certain magical beings can see. It might show a connection between a wizard and a magical creature. In Twilight, it could be something that the Volturi are aware of and use to manipulate relationships. For instance, they could try to sever the red string between a vampire and a human to stop a potential powerful union.