Claim: Katara and Zuko would constantly be at each other's throats and fighting.

Canon: When Katara is mad at Zuko, he lets her be mad at him and doesn't argue back or escalate. He lets her insult him and tell him he looks terrible (as someone who is very insecure about his appearance for obvious reasons), lets her tell him he's terrible at firebending, and in general demonstrates that he's a very safe person to be mad at.

She is the only person he does not raise his voice at or snap at when saying or doing something that irritates him. He raises his voice or snaps at Aan.g, Sokka, and even Toph in one instance for lesser offenses than the times Katara deliberately picks on him. She threatens to murder him and his reaction is just :D -> :c from when he first sees her in his doorway to when he realizes she's mad at him. After she forgives him, we do not see them argue.

The one time they could be said to "argue" it's incredibly brief, impersonal, and resolved immediately upon the misunderstanding being cleared up, literally like two lines of dialogue.

"What's wrong with you?! You could've hurt Aa.ng!"

"What's wrong with me?! What's wrong with all of you?! You're sitting around having beach parties when Sozin's Comet is only three days away!"

(Also note to the "she wants to kill him crowd" that he gave her the exact reason she needed to follow through on the threat to kill him, i.e. "Give me one reason to think you might hurt A.ang...." but all she does is scold him for risking hurting their precious baby boy mutual friend and demands an explanation.)

Claim: A.ang would be the mediator in their relationship and constantly defusing their fights

Canon: The only time anything like this happens, it is Zuko who defuses an argument between Katara and A.ang. Or, in the case of The Southern Raiders, there is an argument between Katara and A.ang and also Sokka, and Zuko takes Katara's side over both of them, stands up for her, and defends her right to seek closure and justice.

If anyone is the arbiter in anyone's relationship, the only canon evidence we can pull from is that Zuko would be the one most likely mediating fights between Katara and A.ang, and standing up for Katara when needed.

In fact, Aa.ng yells at Katara for things not her fault multiple times when they are friends and never apologizes for it. You know who never does that though and is an expert apologizer? I'll give you a hint; it's not Momo.

Claim: Katara hated him / They have a love-hate relationship / They're frenemies

Canon: Are we being for real right now? Did you stop watching after The Western Air Temple? Before she even officially forgives him, who is it that she opens up about her trauma to? Who does she feel safe reliving the worst day of her life with? Who does she cry in front of and who comforts her and tells her that her mother was a brave woman? I'll give you a hint; it's not Momo. And it's not A.ang either. It's Zuko.

She leaps into his arms to hug him at the end of the episode. She sees him block A.ang from sitting next to her and says nothing. After forgiving him, her jabs at him go from mean-spirited to cute friendly teasing. She takes out (what she thinks are) his goddamn baby pictures to show everyone cause she thinks he's so cute. She's the one who encourages him to get in on the group hug (and he does because he low-key will do anything Katara says). She comforts him before he goes to apologize to his Uncle. She doesn't hesitate to go with him to fight Azula. She takes his advice when it's given.

Katara likes Zuko, Katara likes Zuko a lot.

And Zuko likes Katara so much that he acts like a wet dog on Katara's doorstep waiting outside her tent all night for her to wake up because he cares about her feelings so much. Zuko likes Katara so much that if given the choice, he'd trade his own life for hers.