Most likely not. Usually, such stories are made up or exaggerated for entertainment or to create a sense of mystery. There's no concrete evidence to suggest it's real.
Yes, there are some. You can start by looking on popular fanfiction platforms like Archive of Our Own. There might be various stories exploring different relationships and scenarios between Kamitani Hayato and Kashima Ryuuichi.
Mai Senpai was a Japanese novel and anime, also known as the " Index of Forbidden Magic Books ". The story is about a group of young people who have the ability to use magic. They meet and explore the magical world together while solving various problems and challenges. Mai Senpai, one of the female protagonists, was one of the main characters in the story. She was a young witch with powerful magical abilities and was also one of the protagonists. She explored the secrets of the forbidden book catalog together with the other female protagonist, Tama Uejo. After the anime was broadcast in 2009, it was loved by the majority of anime lovers and was widely welcomed in Japan and around the world.
There might also be a rival character. This rival could be competing with Chie Hori in something, like a school competition or a talent show. Their interactions would add drama and tension to the fanfic.
One popular 'chie hori fanfic' story could be about her adventures in a magical world. Maybe she discovers she has special powers and has to use them to save her friends.
Mori Senpai would likely become more openly affectionate. In the series, he was often reserved, but as Haruhi's husband, he would show his love more freely. Haruhi, on the other hand, would probably become more aware of Mori's needs. For example, she might notice when he's tired or stressed and take steps to comfort him.