One common theme is self - discovery. Yuuri often discovers new aspects of himself through his relationships with different characters. For example, in many fanfics, his interactions with Victor help him understand his own capabilities better.
One reason could be some fans' desire to explore darker themes. They might want to see how the character would react in extreme situations. Another reason could be a misinterpretation of the character. Maybe they think Yuuri's insecurities can be taken to an extreme in the form of abuse. However, this is not a healthy or positive way of expanding on the character.
One popular 'yuri on ice yuuri x everyone fanfiction' is 'Yuuri's Multifaceted Bonds'. It explores Yuuri's relationships with different characters in really interesting ways. The author does a great job of showing Yuuri's growth through these interactions.
Yes, there might be some 'Yuri on Ice' fanfiction with Yuuri being abused. However, this is not in line with the general tone of the original anime. The anime shows Yuuri as a determined skater with a lot of support from his family, friends, and coach. Fanfiction like this can make the readers feel uncomfortable as it takes the story in a very different and often negative direction. It's important to note that while fanfiction allows for creativity, it should also respect the core of the original work.
It could be quite an interesting experience. They might be surprised at the different interpretations of their relationships and stories. For example, Viktor might be amused by how some fanfictions exaggerate his charm. Yuri could be either flattered or a bit embarrassed by how his insecurities are explored in some fictions.
There could be a fanfiction where Yuri is kidnapped for ransom. The kidnappers target him because of his fame and the wealth associated with his skating career. His family and friends are distraught but soon gather resources and work with the police to try and rescue him. They face many challenges along the way, like deciphering the kidnappers' demands and finding their hideout.