Ozai definitely fostered whatever negative traits existed in her but if I remember correctly, she was the one that suggested the razing of the Earth Kingdom which Ozai ran with. She never really understood why people rejected her or why what she was doing was bad. She even at an early age was fairly egotistical and I would say that she took her brother being favored over her as an insult to her directly, moreso than her actually craving her mother's love.
I don't think so. Her hallucination at the end of her mother is the most insight we get into their relationship, but the conversation never turns to Zuko. It wasn't about how her mother loved Zuko more than her, it was about how her mother just flat out didn't love her...to which she hallucinates her saying that she did, which implies a very complex percieved relationship. Azula lies to herself, we can agree on that much, if only because of how she obviously doesn't believe she is affected by the betrayals of her friends whne she obviously is. Based on that, my interpretation was that because she saw her mother's doubts in her, she retreated into her father's arms and dedicated herself to emulating him, and just reasoning that her mother didn't love her, when she actually did, but she couldn't handle the complexity of her mother both loving her while still being very much scared of her and her actions.
To the extent that she doesn't understand her actions....that one is hard to say. The smaller examples show she can regret (the Ty Lee example), so we know she CAN connect the dots on a small scale, but she is rarely confronted on her larger actions in general. Who in the fire nation military is going to criticize her actions of capturing the city they've been trying to capture for 100 years? She's usually facing an enemy where she is reasonably encouraged to do all in her power to stop. Which is why part of where she doesn't conform to social norms is arguable. She very much does in her capacity as a warrior princess.
The closest thing she had to an actual relationship was with her father and that isn't fleshed out enough to the point where I can say she actually cared about him as another human being.
You don't need a deep relationship for that. One moment is sufficient. She is under the impression that they will go raze Ba Sing Se together, as a Father-Daughter weekend sort of thing. But Ozai abandons her because he wants to strut around in his new "I'M EMPEROR OF THE WORLD" shirt. To make up for this, he gives her the ENTIRE fire nation. But this doesn't make her happy. She is the second most powerful person in the world, and she isn't happy because her father didn't take her out on their big weekend together.
Related to that slightly, if she truly saw him as nothing more than a stepping stone to power, she wouldn't have lead the avatar and his friends away from him during the day of Black Sun. There was literally nothing for her to lose and everything to gain in that moment. Ozai would have been defeated, the sun would have come back soon, and she would have been the new leader of the fire nation pending Ozai's defeat. In terms of gaining power, this was the most obvious step to take in the world. She might have even gotten the avatar in the aftermath.
Hmmm... I don't know... I couldn't tell you why, but I feel like insofar as Azula admits weakness to Ty Lee in that scene, she's doing it as part of a considered tactic to get what she wants, which is Ty Lee's 'friendship', and she's doing it because that's what she's learned that a friend would do. I mean, if you pressed me I couldn't really justify it, but that's how I always took that scene.
I don't want to push that Azula has true ASD or whatever, but I'm not simply 'writing her off' as a psycho when I call her a sociopath. I mean, to a degree I don't even truly believe she was, like, damaged from birth or something - we don't really see a lot of her childhood (I haven't read The Promise or anything so idk if it's explored there). I just don't think it removes her complexity as a character to call a spade a spade.
No, this is Ty Lee. You could probably set her puppy on fire in front of her and she'd still hang out with you the next day with no ill regard....okay, I'm exaggerating, but you could atleast threaten to set her on fire and she'd do it, because she has. Either way, Azula being mean wouldn't have been sufficient to end a friendship anyway. Keep in mind, if she really pushed, Azula could just force her to do what she wanted by threatening her family or something. In the mindset that Azula claims to have, this would amount to the same thing: Ty Lee doing what she said. But she opts to make up a hurt feeling that, honestly, wouldn't have made any difference in the long run.
Maybe it doesn't for you, but it does for a lot of people. Not everyone has the same definitions of things even that are generally agreed on, so when the discussion for whether she is a sociopath comes up, I always feel obligated to argue against it because even if you don't mean it in the technical sense, people will interpret it as the stereotype of a sociopath, without all the subtleties that make Azula such a fantastic character.