Anon asked: How do you think it would turn out if Azula was sent with Zuko on his quest?
First of all: she'd be absolutely FURIOUS at Ozai. How dare he send her on a stupid quest that he knows it's doomed? And all because Zuko pissed him off? It wouldn't matter that sending her along was the only way to make sure Zuko wouldn't do anything stupid, it wouldn't matter that she would be sent back home eventually, and it wouldn't matter that she was technically the heir now - Ozai betrayed her.
That means the beginning of Zuko's banishment would be even more miserable, because Azula is 100% taking out all of her frustration and pain on him. They'd be fighting A LOT and the crew would find it almost unbearable to be at the same boat with them.
Eventually, however, Zuko and Azula would be forced to work together, and would end up developing a bond (despite refusing to admit it). Neither of them wants to make a fool out of themselves and enrage their father, they're both trying their best for their family and nation, and they were both throw in a different and sometimes hostile environment against their will. They just can't help but turn to each other for help.
That truce would lead to them developing a deeper respect for each other. Zuko would understand that Azula wasn't just "born lucky", and that she actually works really hard to be as powerful as she is, and would be grateful that she isn't trying to sabotage him in any way despite thinking their mission is pointless (even though he is aware she's going along with it mostly for Ozai, not for him).
Azula for her part, would see that her brother isn't weak at all (in any sense of the word). He has quite the temper and is very reckless, but he does have a brain, he can endure a lot (both physically and emotionally), he knows how to take charge AND how to back down when needed, and he is just as dedicated to their nation and their family as she is - and she wouldn't admit it even to herself, but Zuko (and Iroh to a lesser degree) has been treating her better than Ozai ever did, both as the "autority" and as an older family member that is meant to watch over her.
Obviously, during those years away from home, they'd realize that miss what their relationship used to be like, with them being actual friends and always having each other's back. Their feelings would slowly grow into being much, much more than familial.
Once they do find out that the Avatar really is alive, everything changes for them. They'd be one hell of a team, and Azula would prevent Zuko from doing the stupid shit she got up to in book one (like fighting a waterbender in the snow, during a full moon). She'd also finally get a good look at just how competent Zuko can be and how he can improvise when shit gets rough - he always finds a way to track the Avatar (be using rumors, basic logic, or straight up hiring a bounty hunter), he knows how to make deals that benefit him more than everyone else involved (using the very expensive waterbending scroll to make the pirates help him capture Aang - who they have no clue is the Avatar and therefore worth A LOT more), he can stand his ground in a fight (his Agni Kai with Zhao, his sword fight with the pirates), and he can even survive having his ship fucking explode. And he is cunning enough to temporarily go against his own nation's best interests just to make sure HE is the one who will capture the Avatar (the whole blue spirit thing, which doubles as showing off his ninja skills).
Azula probably wouldn't be able to kill Aang in the book 1 finale since it isn't just Aang but also the ocean spirit, making it an unbelievably powerful enemy she cannot defeat. After that, either Ozai will blame just Zuko and Iroh for that failure and make the mission to capture the Avatar be now solely Azula's, or she'd also be a disgrace in his eyes and he'd try to throw her in prison with Zuko and Iroh (key word being TRY).
From then on, two things can happen:
1 - Zuko and Azula will be a dangerous duo (Iroh's is there too, but come on) that will give the Gaang a bunch of problems, then be forced to live as refugees in Ba Sing Se and eventually go back to the palace after the Gaang successfully saves Ba Sing Se from the Dai Li, and invades the Fire Nation in the eclipse, ends the war and has Iroh be Fire Lord (he's a chill, likeable guy, he'd convince them he knows what he is doing). Zuko and Azula would end up growing to understand the mistakes their nation made and work to keep the world at peace... and would one day rule as Fire Lord and Fire Lady.
2 - The events of book 2 would be mostly the same, but Azula and Zuko would have sort of a non-spoken agreement to not get in each other's way, and I can see them almost capturing Aang together in The Chase until Iroh "accidentally" screws up and allows him and his friends to escape. Once Azula knows he is in Ba Sing Se, she gives him the chance to come home with her and he instantly accepts, while Iroh is imprisoned after they take the city and Azula kills Aang. Once they're at the palace, their romance would officially start, but Zuko would still feel guilty for supporting this war and still leave the Fire Nation during the day of the eclipse. Mai and Ty Lee would still save him from being captured at The Boiling Rock (partly due to Mai's feelings for Zuko, and also because they know Azula would regret it if she handed him to Ozai). Instead of a fight "to the death" in the finale that Zuko accepts to prevent anyone else from getting hurt, their Agni Kai would be Azula trying to get revenge on him from leaving her, while he defends himself and refuses to attack her and keeps saying it doesn't have to be this way and that him changing sides doesn't mean he no longer loves her. If we're going for extra drama, Azula still shoots lightning at him, but breaks down crying right mere seconds after it. Once she surrenders/Katara heals him and he sees her sobbing, Zuko embraces her, tells he is going to help her heal, that she is the love of his life and that nothing would ever come between them again, while Azula holds on to him like her life depends on it.