Aang might go into a deep state of depression. He has always been an emotional character, and having his heart broken by Katara, someone he loves deeply, could send him into a downward spiral. He might isolate himself from his friends and stop practicing his bending for a while.
In some fanfiction, Aang's tattoos might be depicted as glowing with a special power when he reaches a new level of understanding or connection to the Avatar state.
Yes, if the title 'aang is a firebender fanfic' is to be taken literally, then aang is likely a firebender within this fanfic. However, without actually reading the fanfic, there could be some sort of plot twist or alternative interpretation that might make it not so straightforward.
One possible reason could be his strong moral and spiritual beliefs. Aang is a very spiritual character, and his culture and upbringing might have instilled in him the value of all life. Killing goes against his principles of peace and harmony.
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.
His refusal can also build tension. Other characters might not understand his stance, or they might be frustrated by it. This can create conflict within the story, both between Aang and his allies, and between Aang and his enemies who might not expect his mercy. For example, some of his friends might think that killing a particularly dangerous enemy is the only way to ensure safety, but Aang stands firm, causing a rift in their relationships. This adds an extra layer of drama to the plot.
Naruto's experience in handling loss and grief can be of great value to Aang. Both have lost important people in their lives. Naruto can show Aang how to turn that pain into strength. By sharing his own journey of dealing with the death of his loved ones and still moving forward, Naruto can help Aang not be consumed by his own losses and instead use them to fuel his growth as the Avatar.
Since 'lemon' fanfic is inappropriate, we can't really discuss the characteristics in that context. But in general, aang is known for his kind - hearted and adventurous spirit. Toph is strong - willed and independent. In proper fanfic, these traits are often explored in their relationship, like how aang's optimism might contrast with toph's more cynical view at times.
He could meet his future self. It would be an interesting encounter as he could learn from his future experiences and wisdom. Or he might meet people who have studied his past adventures and see himself through their eyes.