There are several possibilities. It might be an airbender who was born outside of the Air Nomads, perhaps from a mixed - race family. Or it could be an ancient airbender who was trapped in a spirit realm and has now returned. Another option is that it's a new airbender who has spontaneously developed the ability through some kind of unique event or connection to the spirit world.
Yes, there are likely fanfictions out there with this concept. The idea of a female Naruto in an Avatar: The Last Airbender - inspired setting can lead to some really interesting crossovers. Fanfiction writers often like to mix different universes to create unique stories.
Well, without having read it specifically, I can only guess. But generally in the Airbender world, it's a place full of elemental powers. So this novel might tell the story of the last airbender's struggle. It could be about their fight against the forces that are disrupting the balance in the world. They might be on a quest to find other airbenders who might still be alive, or to master all the airbending skills to become a true savior of their world.
I'd recommend 'Avatar: The Next Generation'. It takes place a few decades after the end of the original series. It follows the new Avatar, who is a waterbender this time, and his adventures. The fanfiction is full of action - packed battles and also delves deep into the new political situations in the four nations.