I'm not sure about the absolute best, but some authors might stand out if they have a deep understanding of both Naruto's and Rin's characters. Those who can capture Naruto's determination and Rin's kindness in their writing would be great. Look for authors who have a lot of positive reviews on fanfiction platforms.
It could be a rogue ninja from a hidden village who has a vendetta against Naruto for some reason. Maybe Naruto unknowingly disrupted this ninja's plans in the past, and now the ninja wants revenge. This would set up a situation where Naruto is in danger and needs Rin's help to be rescued.
The story could also focus on the family dynamic. Naruto, being a very determined and positive character, could be a great father figure to Rin. He could encourage Rin to face his inner demons (both literally and figuratively) with the same kind of courage that Naruto has. And perhaps Naruto would have to protect Rin from the more dangerous aspects of the Blue Exorcist world using his ninja abilities.
In a fanfiction world where Naruto and Rin have a baby, it could be a very interesting story. Maybe the baby would inherit Naruto's positive attitude and Rin's kindness. The story could explore how they raise the child in a world full of challenges and ninjas.
Not really. Most Naruto fans love the original characters and their relationships. Bashing a character like Rin might not sit well with a large portion of the fan base. So, these types of fanfics are likely not as popular as those that stay true to the positive aspects of the Naruto characters.
Kakashi is known for his laid - back yet highly skilled nature. His influence on Naruto could be in terms of teaching him to be more strategic in battles. Rin, being kind - hearted and having a strong sense of care, would give Naruto the emotional support he lacks. So Naruto might become more balanced in his approach to life and ninja - hood.
There could be other Vocaloid characters. For instance, her brother Kagamine Len might be a main character. They often have a special bond and their interactions can be a significant part of the story.
There are indeed such fanfictions. In these stories, Rin's survival can have a huge impact on the other characters. For example, it could change Obito's path completely. Instead of becoming consumed by hatred, he might have a different future with Rin alive. Also, Kakashi's character development could take a different turn as well, since Rin was an important part of his past.
If Naruto and Rin had a baby in fanfiction, it might have unique abilities. Since Naruto has the Nine - Tails' power and Rin was a skilled kunoichi, the baby could inherit some extraordinary skills. For example, it could have a special form of chakra control.