In fanfiction, the Rinnegan is often depicted as more powerful than the Sharingan in many ways. The Rinnegan has abilities like the Six Paths techniques which can be extremely overpowered. For example, it can allow the user to control gravity, summon large creatures, and even revive the dead. While the Sharingan has its own set of cool abilities like predicting movements and casting genjutsu, it seems limited in comparison when it comes to the grand scale of power shown by the Rinnegan in most fanfictions.
Naruto would probably win. He has an indomitable will and has always found ways to overcome powerful foes. His experience and the unique powers he might gain with the Rinnegan could give him an edge over Madara.
Their relationship would likely become more complex. With the Rinnegan involved, they might have to rely on each other more for understanding its powers. They could become more protective of each other, knowing the power and the danger it brings.
One implication is that he becomes much more powerful. The Rinnegan gives him access to powerful abilities like the Six Paths techniques, which would change the power dynamics in battles in the fanfiction. He could now face stronger enemies more easily.
In many fanfictions, the Rinnegan and chakra chains are often intertwined in terms of power and function. The Rinnegan, being one of the most powerful ocular abilities in the Naruto universe, can potentially bestow special properties on the chakra chains. The chakra chains themselves might be a way to channel the raw power of the Rinnegan. For instance, they could be used to bind opponents while simultaneously draining their chakra through the Rinnegan's absorptive capabilities. This connection adds an interesting dynamic to the characters who possess both the Rinnegan and the ability to use chakra chains, making for some really cool and unique battle scenarios in fanfiction.
Yes, there are likely to be 'harry has the rinnegan' fanfictions out there. The idea of combining elements from different fictional universes, like Harry Potter and Naruto (where the Rinnegan comes from), is a popular concept in fanfiction. Writers often like to experiment with giving characters from one world the powers or traits of another.
Another possibility is that it's a result of some strange chakra experiment gone wrong (or right). Scientists in the Naruto universe are always dabbling in chakra research. Maybe an experiment meant to enhance her natural abilities instead gives her the Rinnegan, and she then has to learn to control it quickly to avoid being consumed by its power.
If Naruto got the Rinnegan, he might become more cautious. The power of the Rinnegan is so great that he would realize he needs to be more careful with it. He wouldn't be as brash as before.
It could be Sasuke. If he were to turn to the dark side in the fanfiction, his Rinnegan powers could be used to destroy Konoha. He has the potential to be extremely powerful and if he has some sort of vendetta or is influenced by some dark force, he might take his anger out on the village.