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So Azula wins the Agni Kai. Katara entering the Arena gives Azula the right to attack her. She burned Zuko winning the Agni Kai.
However she is imprisoned and mistreated e.g beatings. Iroh pressures Aang into taking away her bending so she is no longer a threat. Believing Iroh cannot be wrong and other people in the Gaang supporting it Zuko and sokka he agrees. This is Aang worst mistake as energybending requires a pure spirit and conviction. Ravaa realising Aang was going to fail to do it and end the Avatar cycle.
Offers to send Azula back in time just after ba sing se in order for her to become the Avatars firebending teacher and the chosen fire lord. As Zuko leadership as Fire Lord was doomed to fail as he is a disgraced traitor who had been disherited and is working with the enemy.
Azula does not fully agree with the idea of the Fire Nation Accepting guilt but see peace as the only way forward. She was not feed the same properganda as heir so know the truth about the Air nomad and thought it a foolish mistake but it could not be un done. However, she will not give up the original colonisers made whilst roku was still alive. Head canon the original colonise were never earth king territory but what was left of the chin dynasty otherwise it would have started al out war during the time of roku and they have been assimilated very well over the last 125 years.
Azula being Aangs Fire bending teacher mean that he learns that killing is some times nessary so he never learns energy bending. They still go and see the dragons. Zuko choose to follow his Uncle instead of the Gaang.
Azula becomes Fire Lord unopposed as Zuko is helping Iroh free Ba Sing Se. Iroh tries to pressure Zuko into challenge Azula throne but Zuko hears the news that Iroh tries to hide from him that Azula taught the Avatar and helped down the Air fleet and point out Iroh's hypocrisy that
He said two brother fighting for a throne is an act of senseless violence.
Azula has a Tyzula and Zuko Has Maiko and Zuko. The search happens dismiss the whole ursa changes faces and forgets her children. Instead their are angry spirits who are protecting dragon eggs and a dying dragon from hunters. And Ursa being to poor to return home and their being no offical end to her banishment as she was never officially banished in the first place