So, question, because I see people saying it often that Iroh has the right to feel animosity towards Azula because she made fun of/derided Lu Ten's death (or something like that), but does she actually do that in that scene? Or does she express disdain for Iroh's reaction, which considering the culture could easily be interpreted as Iroh being the one to make light of it? (Pretty sure there's at least one instance in _Romance of the Three Kingdoms_ where one character absolutely annihilates an opposing force because his brother/father/friend dies, if we want a real-world example of the mentality. Or, like, all of _The Hagakure_.)
Does Azula call Lu Ten a coward for dying? Or does she say that Iroh is for not "getting justice" or revenge for his son's death? For not finishing the task and abandoning the cause Lu Ten died for?
Because one of these means Iroh's dislike could be justified (nevermind the fact that he wasn't present for this conversation, so if he knew about it, he would have only heard about it from Zuko). But the other is an angry/disappointed/disgusted child calling an adult out.
Good question! I think I should start by talking about what Zuko and Azula actually say about Iroh.
"The Western Air Temple"(featuring 13 year old Zuko!):
Iroh: (Iroh looks on, concerned) Prince Zuko, it's only been a week since your banishment. (Cut to a far back view shot of the 2) You should take some time to heal and rest.
Zuko: (turns around and raises his voice) What else would I expect to hear from the laziest man in the Fire Nation? (Cut to a close up of Iroh's slightly appalled face as he looks down and sighs) The only way (Cut back to a frontal shot of uncle and nephew) to regain my honor is to find the Avatar. So I will.
Zuko: (standing at the bars) You brought this on yourself, you know. We could have returned together. You could have been a hero! (Iroh turns a shade further away from Zuko.) You have no right to judge me Uncle. I did what I had to do in Ba Sing Se, and you're a fool for not joining me. (Iroh is silent.) You're not gonna say anything? (Enraged, he kicks a stool and bends a blast of fire at the wall.) Argh! You're a crazy old man! You're crazy, and if you weren't in jail, you'd be sleeping in a gutter!
Zuko says some pretty negative things about Iroh, right to Iroh's face!
Now, what negative things does 14 year old Azula say about Iroh? Surprisingly little, even though she clearly doesn't like him. She implicitly calls him a traitor a couple times(during times when he is, in fact, a traitor by all reasonable definitions), but never really explicitly does so. Beyond that, there's very little. This is the only thing I can think of:
Azula: So...I hear you've been to visit your Uncle Fatso in the
prison tower.
Zuko: (standing, incensed) That guard told you.
Which is actually way less harsh than what Zuko says about Iroh! If anything, Azula's behavior in the present suggests that she only rarely criticized, much less mocked Iroh's behavior to his face when she was younger.
Now let's turn to the meat of your question, "Zuko Alone." There are two scenes in that episode where Azula criticizes Iroh. The first comes before Lu Ten's death:
Ursa: "And for Azula, a new friend. She wears the latest fashion for Earth Kingdom girls."
(As Ursa speaks, Azula picks up a doll wearing Earth Kingdom green. The Princess makes a face of disgust.)
Azula: If Uncle doesn't make it back from war, then dad would be next in line to be Fire Lord, wouldn't he?
(In the background, Zuko runs around practicing with his new dagger.)
Ursa: (disappointed) Azula, we don't speak that way. It would be awful if Uncle Iroh didn't return. And besides, Fire Lord Azulon is a picture of health.
Zuko: How would you like it if cousin Lu Ten wanted dad to die?
Azula: I still think our dad would make a much better Fire Lord than (looking at the doll with disdain) his royal tea loving kookiness.
(She holds out the doll and makes its head burst into flame. The screen flashes white and the flashback ends.
There are several things which seem to be driving Azula's actions here. The first is a reaction to the massive favoritism Iroh just showed toward Zuko. The second is a belief, no doubt inspired by Ozai's poisonous statements about his brother, that Ozai, who Azula idolizes, would make a better Firelord than Iroh. Finally, Azula is a confused child who is asking inappropriate questions because she's too young to understand proper boundaries. Nothing she says here is actually that serious, and I would expect a responsible adult(i.e. not Ursa) to either shrug it off, or to carefully reason with Azula in order to explain why what she is saying is problematic.
Now we turn to the other main scene, the one right after Lu Ten's death, and the one you probably actually wanted me to talk about:
Azula: (getting up and walking over to him) By the way, Uncle's coming home.
Zuko: Does that mean we won the war?
Azula: No. It mean's Uncle's a quitter and a loser.
Zuko: What are you talking about? Uncle's not a quitter.
Azula: Oh yes, he is. He found out his son died and he just fell apart. (leaning against a nearby pillar) A real general would stay and burn Ba Sing Se to the ground, not lose the battle and come home crying.
Zuko: (angry) How do you know what he should do? (looking down, sadly) He's probably just sad his only kid is gone... forever.
You might note that, again, Azula doesn't say anything negative about Lu Ten. You've already noted that Iroh is thousands of miles away at the moment, so having him be "justified" in his hatred of Azula by a conversation he didn't hear doesn't make sense.
Azula is also very angry in this scene, quite possibly the angriest we ever see her at any point. That does suggest that she's taking what's happened, either Lu Ten's death or the abandonment of the siege, very personally. I don't know enough to comment specifically on this, but you are right in that there might be cultural background which specifically proscribes the achievement of vengeance as being of particular importance, and Azula is thus outraged that Iroh failed to fulfill his duty to his son.
But that not at the core of Azula's critique here. What Azula is attacking Iroh for is that he responded to a personal loss by abandoning his duty in the heat of battle, and she is 100% right on this, not only by the standards of Fire Nation cultural but also by the standards of modern western culture. Fun fact: Abraham Lincoln, Jefferson Davis, and Robert E. Lee(screw the latter two, but that's another story) all suffered the loss of children they dearly loved in the middle of the American Civil War, yet none of them abandoned their duty. Archibald Roosevelt(another problematic figure) had two brothers die in WWII, yet he continued fighting on the front line. John W. Geary literally had his son die in his arms in the middle of battle, yet he continued commanding his unit well enough to prevail. Hell, we can even turn to Joseph Stalin here, to some extent.
Again, the core of what Azula says here is absolutely correct. Iroh is a "quitter" because he responded to Lu Ten's death by falling apart and abandoning the siege when it seemed on the brink of success, rather than continue the operation until victory. I don't think we need to go further than that to establish that Azula is entirely justified calling Iroh out here. And again, she doesn't criticize or mock Lu Ten at all, instead only attacking Iroh's reaction to Lu Ten's death.
