Rip my heart out why don’t you.
1. Katara would kill Azula. Katara’s anger burns very close to the surface, and not only does she tried to kill Katara, and succeeds in killing Zuko, which itself is an emotional blow, but she has done so that brings to mind how she killed Aang. Besides this, she has just proven how dangerous she is again. I think Katara would kill her. When she freezes her, she just leaves Azula that way to suffocate.
2. I go back and forth over whether Iroh would take the throne if Zuko were unavailable. In this case, I think he would, because he would want someone he knew very well leading his home country, now that two of their royals died so close together. So he would take the throne, devastated at the lost of his second child, and of his niece.
3. Because Iroh is an old childless widower, he would have to either appoint an heir from within his family, and right now family is thin on the ground, or remarry a younger woman and try to have children with her. This leads to a search for a suitable woman. He might settle on Mai, who would be mourning Zuko as well, so at least they share that, but I think he would try to look for an older woman, by which I mean older than Mai, in her twenties or thirties , who has been known for her political opposition to Ozai.
4. Iroh ascending the throne would be diplomatically very tricky, because not only is he in mourning again, but also, he is a war criminal and a boogy man to the Earth Kingdom. Aang and the rest would have to work hard at smoothing this over and pushing the changed man view of The Dragon of the West.
5. Because Aang was able to spare Ozai and Katara killed Azula, Aang would probably feel a certain level of disapproval of Katara’s actions. Their differing morals with regards to killing would be something they would have to work through. I think they could though.
I don’t think Katara would have guts to end Azula. She most likely
would injure her to the point that she make Azula disabled, but not kill
her. If she spared man that killed her mother she would spare Azula
too, but at that point Azula most likely would have her arms and legs
broken. If she kill her she most likely would have serious emotional
trauma. It would hinder greatly her relations with Aang.I
also don’t think Iroh would take Mai (its your fan fantasies) as her
wife after Zuko’s death. She is too young for him, also her father is
known Ozai supporter. So in fear that Ozai could return to power he
would choose something else.Being unable to go through with a premeditated revenge murder is not the same as being unable or unwilling to kill in the heat of battle someone who has already proven to be an extreme danger, and who has just killed a friend of yours in front of you. Those are very different things that require very different mental capabilities. Actually disabling someone the way you describe takes more planning and forethought, and more calculation. If anything, Katara might find that harder in the moment. Afterwards, killing Azula may well be extremely traumatic for Katara, and would definitely stress her relationship with Aang. Dealing with that side of herself might be very difficult. That doesn’t mean I don’t think that in the moment she would do it.
Calling something my fan fantasies is insulting in the extreme, which I’m sure you know. I mentioned it as a possibility, not because either Iroh or Mai would care for the idea, but because it might be the most politically savvy option. Mai has court connections through her parents, through Azula, and through her own time spent at court as a companion of the princess. She is a noble, and she has been to prison for opposing Azula, which can be spun as protesting the conquest. Also, Iroh is old and has to face the very real possibility that he could die before a child of his is old enough to take the throne. So he has to look for a spouse who isn’t just a suitable wife, politically speaking. A much bigger consideration for Iroh is would she make a suitable regent while any child they had was under age. He would, in short, be looking for a successor much more than a wife. Mai is a possibility for this. However, as I said in my initial headcanon, I think it’s more likely he would pick someone else with similar political pluses, but whom Iroh has a better idea of their views. He may even be looking among other nobly born political prisoners for his spouse, i.e. someone in Mai’s situation who is not Mai, who is older, who he is more certain of, and who doesn’t have the connection to Zuko. Again, calling something my fan fantasies is a lovely way of denying I have thought about it any more than necessary to decide it appeals to my id, which it doesn’t, as it happens. It’s insulting and dismissive.