Yes, in many fanfictions, Naruto can still levitate. Fanfiction often takes creative liberties with the original abilities of characters. Some authors might keep his levitation ability for various plot reasons, like for a new battle technique or to show his growth in a different way compared to the original story.
Mostly, yes. Fanfiction writers love to play with the idea of Naruto's Rasengan. They might give it new forms, like a lightning - infused Rasengan or a Rasengan that can split into multiple smaller ones. Since the Rasengan is so central to Naruto's character in the original story, it often makes an appearance in fanfiction, but of course, there are always exceptions depending on the unique ideas of each writer.
Maybe the author wants to explore more of Naruto's growth at the genin level. There could be many untold stories and character developments that can happen during this stage. It gives the author more space to show Naruto's relationships with his friends, like Sasuke and Sakura, in a different light while still being a genin.
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.
Yes, he could. He could combine his Saiyan powers with ninja skills. For example, he could use his super speed to perform faster ninjutsu. His Saiyan strength could also be used in close - combat situations during ninja battles.
One such book could be 'The Levitating Children of Zephyr City'. It tells the story of a group of kids who discover their ability to levitate and the adventures that follow.
One consequence could be that he might face more discrimination from the village elders. In the original story, as Naruto advanced in rank, he gained more respect. But as a genin here, he could still be seen as a troublemaker. Another consequence is that his training might be different. He might focus more on basic genin - level skills, which could make him even stronger in the long run as he masters them to a higher degree.
Yes, it's possible. In fanfiction, the author can write it so that Naruto's strong sense of self and his values prevent him from being completely overrun by the Gedo Mazo's influence. His bonds with his friends and his love for the village can act as anchors.
In a fanfiction where Naruto's parents are alive, they could play a major role in changing the politics of the Hidden Leaf. Minato, being a former Hokage, and Kushina with her strong Uzumaki blood, might work together to strengthen the village's defenses and relations with other villages. This would also mean Naruto doesn't have to struggle alone and could learn advanced techniques much earlier from his parents.