One popular 'Aang Saves Katara' fanfiction can be found on Archive of Our Own. Just search for the exact phrase and you'll likely get a good selection. There are often stories where Aang uses his airbending skills in creative ways to rescue Katara from various dangerous situations.
A common plot element is that Katara is usually in some sort of immediate danger. It could be she's captured by the enemy, or about to be harmed by a natural disaster. And Aang, with his heroic nature, feels compelled to save her.
Their relationship could deepen as they share in the joys and sorrows of rebuilding the world. They've been through a lot during the war, and now they can truly rely on each other. They might become even closer friends at first, with a mutual respect for each other's abilities growing stronger.
In some cases, the anger could be a catalyst for a more romantic development. Aang expressing his strong emotions might make Katara realize the depth of his feelings for her. Or it could be the other way around, Katara's reaction to his anger could make Aang see her in a new light, leading to a shift in their relationship dynamic towards a more romantic one.
Katara could be heartbroken. She might initially be in denial, not believing that Aang has actually left. Then she could become angry, feeling betrayed by Aang's sudden departure.
It could also be that Katara was being overly critical of Aang's progress in learning new skills. Aang might feel frustrated and angry if Katara is constantly pointing out his flaws instead of being more encouraging, especially if he's already trying his best.
No. Fanfiction that involves sexual content about characters is not appropriate. These are fictional characters often aimed at a wide range of audiences, including younger viewers. Such fanfiction goes against the family - friendly nature of the original works and can also be considered disrespectful to the creators' intentions.
Yes, there might be. There are many fanfic communities where fans write about various scenarios for different fictional characters. You can start by looking on popular fanfic platforms like Archive of Our Own (AO3). They often have a wide range of stories based on different shows and characters, and there's a good chance you'll find one about Aang and Katara in this situation.
They could embark on a quest to find a lost relic. Aang's airbending skills and Katara's waterbending would be crucial in overcoming the obstacles like ancient traps and dangerous creatures. Their teamwork would strengthen their bond.