Saitama might be really frustrated with Mosquito Girl. Since he can defeat most powerful foes with one punch, but a mosquito girl would be a different kind of annoyance. She could be constantly buzzing around and trying to bite him, and Saitama would probably just try to swat her away like a normal mosquito at first. But when he realizes she's a 'villain' in her own right, he might just give her one quick punch to end the nuisance.
In a fanfiction, Mosquito Girl could develop a grudging respect for Saitama. After being defeated by him, she realizes that there are beings far more powerful than her. And Saitama, on the other hand, might start to see Mosquito Girl as more than just a villain. Maybe he discovers that she has some good qualities, like her tenacity. So their relationship could shift from pure enmity to a more complex dynamic of mutual understanding.
Some fanfics focus on Mosquito Girl's power evolution. They might explore what would happen if she somehow managed to gain more power, perhaps through some new mutation or training. This could lead to really interesting battles and power - play situations within the fan - created storylines.
Well, sometimes there are original characters that are created to be Mosquito Girl's allies. For example, a scientist who has a special bond with her and helps her control her powers better. This character could add a whole new dynamic to the story.
Yes, there likely is. The Naruto and One Punch Man fandoms are both quite large, and it's common for fans to create crossover fanfictions. These crossovers can be really interesting as they combine the unique elements of both universes, like Naruto's ninja skills with Saitama's one - punch power.
I'm quite sure there exists a 'Naruto is One Punch Man' fanfic. The world of fanfiction is vast and full of all kinds of creative crossovers. Writers might be intrigued by the idea of bringing Naruto's ninja skills and the unique world of One Punch Man together. For example, how would Naruto react to the overpowered villains in One Punch Man's world? And how would Saitama view Naruto's determination to protect his village? These are the kind of things that could be explored in such a fanfic.