My version of Avatar The Last Airbender backstory:
I think I am going to blow everyone's minds away...
Azulon wanted Ozai to feel the pain of losing a child. But it wasn't Zuko he wanted killed, it was Azula. In the hierarchy of Avatar The Last Airbender, and in historical politics, male figures traditionally dominate. A female was possibly never intended to inherit the throne of the Fire Nation, only males. Still, Azulon knew that Azula was Ozai's favorite, and she would hurt him the most, while Zuko the male would be alive to justify Ozai's lineage over Iroh's. Azulon in turn could not hurt Zuko, or his whole family's place on the throne would be sacrificed.
Ozai began to hate Zuko even then because his prized fire bender, his true kin, was being condemned. Both Ursa and Ozai didn't want to lose their child, though for different reasons. So they conspired and figured out a way to kill Azulon.
Part of their condition to work together, Ozai would set Ursa free. However, he did so because he, being the future Fire Lord and son of the deceased king, knew who placed the poison. He could not keep her around as she could be a liability to his reign. Imprisoning her would also inadvertently acknowledge his guilt under suspicion. The only resolve was banishment. They both get what they want. Though Ursa was tricked as she believed she could leave with her children, but Ozai denies her. Meanwhile Zuko remains at Ozai's side as a means to justify his rule. Ozai ultimately knew that he could never kill Zuko until supreme power was his.
This all is also why Ozai has little care for Zuko when he challenges him to Agni Kai, but only scars him. He eventually entrust the whole of the Fire Nation to Azula and proclaims her Fire Lord. Not quite because his son is a "traitor", but because his intentions was to always have Azula follow in his role as she was the one that resembled his demeanor the most and would carry out his vision.
As for Zuko being his son or not is still left to be discovered in the Search pts 2 & 3. I would write in as such:
Zuko is Ozai's son, but after reading the letter (spoiler), Ozai confronted Ursa and never trusted her again or Zuko's kinship. Ursa and Ikem planned to have children and created names, one of which was for their first born to be named Zuko. She always looked at Zuko as a reminder of her love for Ikem. She would tell Ozai this, though he never accepted her words.
Whether Zuko was his or not, he could not lose his son or heir apparent, though at this period in time Iroh's son is still alive. However, Ozai's conniving self would eventually plan to have Iroh's son killed during the Siege of Ba Sing Sei, giving him leverage to make his proposition to Azulon.
Ozai in secret sends soldiers and bounty hunters after Ikem, but never finds him. He is anguished by the conflict of never knowing the truth of Zuko, and therefore he is henceforth treated indifferently. In the future, Ozai truthfully scars Zuko during their Agni Kai out of rage for Azulon wanting to kill Azula, for not knowing if he is his son, for Ursa's betrayal, for her banishment, and for having so much power and not the ability to control everything even his son. The scar Ozai carried within was transferred to Zuko.
As for Ursa, she returns to her home of origin sadden by her separation from her son and daughter. In time, she would set out to find Ikem, yet, never physically finds him. Ozai sends spies for years to see if Ursa and Ikem encounter, and they come back empty handed. Ursa lives out her life in solace while she thinks of her children. Meanwhile, Azula hates her mother for leaving, but never learns the truth that she sacrificed her life to save hers.
Zuko and Azula eventually discover the truth of their mother, and Azula finds herself changing. She remains in Hira'a to live out a peaceful life.
Zuko returns to the Fire Nation, and tells Iroh the truth about his son. Iroh enraged confronts his brother in his cell. They exchange anger, but Iroh then calms down as he sees Zuko near, and knows that Ozai wanted it all but in fact lost his whole family, even Azula. Iroh never speaks to Ozai again. Iroh is sadden by his family's tragedy. Zuko, kneels to his uncle and begs him to find the will to look at him as his son as he too has lost his father. They embrace and Zuko declares Iroh in front of the Fire Nation as his adoptive father and bestows upon him the title of Dragon Lord. And they all drink tea in his honor.
SPOILERS !!!!!!!!!!!!I like what you hah to say a lot and agree with nearly everything...escpecially the Azula stuff. BUT, I think they will actually find Ikem and Ursa somehow alive. My original idea was that somehow they escaped to the spirit world. Im trying to figure out the significance of that wolf in part 1.
Plus, Ozai confirmed what Azula told Zuko as a child was true: Azulon did order him to kill his son so that he can share his brother Iroh's pain of losing his only son. Ozai actually was going to carry it out. But Ursa formulated a plan to make him Fire Lord and spare Zuko. Unknown events result in Azulon's death and Ozai's ascension to ruler of the Fire Nation.
"Azula eavesdropped on the entire meeting between Azulan and Ozai, she would have known if killing her was the condition proposed."
We all Azula is a psychopath and compulsive liar. Even at an early age she could have developed a will for survival. Also, by this period she was already bullying Zuko. She could have heard part of the conversation Ozai and Azulon had, but heard what she wanted to hear. Upon the order, her defense mechanism kicked in, and she turned to her patsy. It could be that something psychological triggered in her mind to say what she said to Zuko. She wasn't empathetic, but malicious over for the news.