In many fem Aang and Zuko fanfictions, the 'soulmate' trope is present. It gives a sense of destiny to their relationship. The 'secret relationship' trope is also quite common, especially considering their different backgrounds and the political implications. They have to keep their relationship a secret at first. And the 'training together' trope, where Zuko helps fem Aang improve her firebending skills while their relationship deepens during the training sessions.
A common trope is the 'redemption arc' for Zuko within the context of his relationship with fem Aang. It often involves him trying to make up for his past actions while falling in love with her. Also, the 'cultural exchange' trope is popular, with fem Aang introducing Air Nomad culture to Zuko and the Fire Nation, and Zuko teaching her about Fire Nation traditions. There's also the 'forced proximity' trope, like they have to work together closely on a mission and feelings start to develop.
One popular trope is 'enemies to lovers'. Since they started as enemies in the show, seeing their relationship develop into love in fanfiction is really appealing. Another is 'forbidden love', as there are elements in the Avatar world that could make their relationship difficult.
One idea could be a time - travel fanfiction. Zuko and Aang could be sent back in time to the era before the war started. They would then have to use their knowledge of the future to prevent the war from ever happening. This would involve them working together to change the minds of key people and stop the events that led to the conflict.
One popular plotline is Aang and Zuko teaming up to face a new threat to the Four Nations. Maybe a long - lost ancient evil that requires both the Avatar's bending skills and Zuko's strategic mind. Another is a story where they go on a journey to understand each other's pasts better. This could involve visiting places that were significant in their respective childhoods.
You can try fanfiction websites like Archive of Our Own (AO3). There are a ton of fem Aang and Zuko fanfictions there. Just use the search function and type in 'fem Aang Zuko' and you'll get a list of relevant stories. Another option is FanFiction.net, though the tagging system might not be as precise as AO3.
Maybe Zuko has his own goals that don't involve chasing Aang. He could be more focused on redeeming himself in other ways, like helping the Fire Nation people or learning about different cultures without the need to pursue Aang.
One popular trope is the 'enemies to friends' trope. Initially, they were on opposite sides, with Zuko as the pursuer and Katara having her reasons to be wary of him. But as the story progresses, they overcome their differences. Another is the 'redemption arc' where Zuko's journey towards redemption is intertwined with his relationship with Katara. His actions to make amends often involve her in some way.
One popular trope is the 'Chosen One' aspect. In fem Harry's case, she could be chosen by Poseidon for a special quest. Another is the 'Mentor - Protégé' relationship, where Poseidon guides fem Harry in her journey, perhaps teaching her about her newfound powers related to the sea. Also, the 'Forbidden Love' trope can be used, as their different natures - a human - like fem Harry and a god like Poseidon - might make their relationship something that others oppose.
A 'secret relationship' trope can be found in many of these fanfictions. Given Thorin's position as a leader and the differences in their backgrounds, they might have to keep their relationship a secret at first. Also, the 'soulmate' trope is popular, where they are somehow fated to be together, which can create a really romantic and engaging story.
One common theme is friendship. Since they start off as enemies, many fanfictions focus on how they build a strong bond of friendship. Another is redemption. Zuko's journey of redemption is often explored further in relation to Aang.
I'm not sure specifically as there could be many different stories within that fanfiction. It might be about Zuko and Aang's relationship, perhaps in an alternate universe or a situation not shown in the original series. It could explore their friendship, their adventures together, or even some creative scenarios that fans have come up with.