Yes, in the vast world of fanfiction, there could be such a concept. However, Naruto and Sesshomaru are from two very different anime universes, Naruto being from the ninja - filled world of 'Naruto' and Sesshomaru from 'Inuyasha'. But fanfiction writers often like to cross boundaries and create unique and unexpected relationships. They might come up with elaborate backstories on how such a thing could be possible, perhaps involving some sort of magical or time - traveling plot device.
One possible plot is that Naruto, as Sesshomaru's son, discovers his true identity later in life. He then has to learn about his father's world and culture. There could be a conflict between his loyalty to Konoha and his newfound family ties to the demon world. He might also encounter other demons who either want to use him for his power or are out to get him because of his mixed heritage.
Naruto's character would change a lot. He'd become more reserved perhaps, as Sesshomaru is a rather aloof character. But he'd also have a new sense of power and pride. His development could involve learning to control his new demon powers, which might be more difficult than his chakra control. He might start to view the world from a more demon - like perspective, seeing things in terms of power hierarchies more often.
In such a fanfiction, there could be a lot of cultural clashes. Naruto is used to the shinobi way of life, with its clans and ninja villages. Sesshomaru's world in 'Inuyasha' has its own set of rules regarding demons and humans. Naruto might try to bring some of his ninja ethics to this new world. There could also be a plot where Naruto has to prove himself to Sesshomaru's followers or enemies. His Naruto - style combat techniques would be a novelty in this world, and he could use them to his advantage while also learning new skills from his father and the other characters in the 'Inuyasha' world.
Sesshomaru would likely be aloof and haughty at first. He might look down on Naruto as a mere human. Naruto, being his friendly self, would probably be curious about Sesshomaru and try to start a conversation. Sesshomaru might respond with cold remarks, but Naruto's persistence could gradually break through his icy exterior.
There are definitely some out there. I've seen ones where Naruto is put in a situation similar to Sesshomaru's, like having to deal with a powerful enemy or a complex moral dilemma in a way that Sesshomaru might. It's really interesting to see how authors blend the two very different universes of Naruto and Inuyasha (where Sesshomaru is from).
There probably is. Fanfiction writers often come up with very creative and unexpected crossovers. However, the quality can vary greatly. Some might be really well - written with interesting plotlines that blend the Naruto and Inuyasha universes in a unique way considering the relationship you mentioned, while others might not be as good.
One possibility could be that in this fanfiction, female Sesshomaru uses her demon powers to protect Naruto. Maybe she has her own motives, like she sees a kind of potential in Naruto similar to what she once saw in someone else. She could be a very strict but effective guardian, teaching Naruto new combat techniques from her own demon - related knowledge.
One common theme is power. Since both Naruto and Sesshomaru are powerful in their own ways, fanfictions often explore how Naruto would handle Sesshomaru - like powers or how he would adapt to new abilities. Another theme is identity. Naruto has his own strong identity as a ninja, and when combined with Sesshomaru - like traits, it creates an interesting exploration of who he becomes.