I have a suspicion that Ozai was sexually abusing Azula....
I don't like that theory for one simple reason (aside from the 'ick' factor) it totally absolves her of all responsibility.
Not at all. Responsibility is not so simple as that. We're all shaped by our life experience, for better or worse, and that influences what we become, but we are still ultimately culpable for our own choices and actions. It's the way of the world that victims become victimizers, so you can't draw a clear dividing line between the two. Recognizing that someone wasn't "born evil" but made that way by a traumatic past doesn't absolve them and let them walk away scot-free; it just affords understanding of how evil happens and what we can do to prevent other people from turning out the same way, and possibly to find the key to rehabilitating wrongdoers so they cease to be a threat.
After all, Zuko was overtly abused by his father, though not sexually. He bears the proof of that abuse right there on his face. Indeed, more than half his family was abusive toward him; he was scorned and belittled by his father and tormented by his sister, and his own grandfather wanted to kill him. So he was also shaped by the traumas inflicted on him by his family. But he overcame the damage that did to him and turned away from the darkness. Azula didn't. That's where individual culpability comes in.
I prefer to think that Iroh's assessment of her is the most astute; "she's crazy and she needs to go down." After the lengths he went to trying to get Zuko to unlearn what his brother had instilled in him and from what we've seen of his character in general, I hardly think he'd 1) fail to notice his neice has been abused and 2) neglect her.
When the series starts, Iroh's been away from court for, what, three years? That means Azula would've been 11 the last time he saw her before season 2. And he'd been away at war and lost his own son just three years before that. Even if the abuse started that early, he might've been too caught up in his own grief for a time, and would probably have been pretty marginalized from the royal court, given that he was the rightful heir to the throne and got shoved aside by Ozai. So he may not have had much interaction with Azula as she grew up. Heck, if he had, she would've turned out better, don't you think?
Some kids are just born not right in the head. If they happen to be raised in a climate of power and control (with a genetic history full of mudering nutters) then it's hardly surprising you get the odd sociopath.
Well, sure, but it's a mix of both. Inborn nature is just potential; how it's realized depends on upbringing. A child who's born with a deficiency in the capacity for empathy and social development, if raised in an abusive or neglectful environment, may well turn out cold, uncaring, sociopathic, even homicidal. But if the same child is raised in a loving, nurturing, stimulating environment, it will cancel out that deficiency, in the same way that an intrinsic physical weakness can be compensated for by vigorous exercise and training. Stimulation promotes development, neglect promotes atrophy. Heredity and environment must both be taken into account.
As for those who think her breakdown was too fast, as I recall her very first appearance showed her to have an obsession with perfection to the point where a single hair out of place sent her into a rage.
Well, technically her first appearance is in the opening titles of the pilot episode; she's the firebender demonstrating "Fire" in the opening shots. And her first in-show appearance was a brief cameo at the end of the first-season finale. But that's, uh, splitting hairs.

You're referring to a scene in the season-2 opener:
(Cut to Azula's battle ship and her lightning bending practice. The sun is setting in the background. Azula stands on the deck and brings her arms down in front of her. Cut to two old women watching from the top of the stairs.)
(Azula swings an arm around, electricity sparking from her left hand. She brings it around in a circle, then begins to mimic the motion with her right hand, electricity shooting from her index and middle fingers. She brings her fingers together, and then lunges forward, launching a lightning bolt from her left hand. The lightning arcs over the edge of the ship. Cut to in front of Azula, her fingers smoking and a single strand of hair dangling in front of her face. Cut to the old women on the stairs.)
Elder 1: Almost perfect.
Elder 2: One hair out of place.
(Cut to the same scene of Azula. The camera zooms in quickly as Azula glares angrily at the one strand of hair. She wipes it from her eyes.)
Azula: 'Almost' isn't good enough!
(She begins the training again and shoots lightning at the camera.)
http://www.avatarspiritmedia.net/transcripts.php?ep=201
I saw that as simply the perfectionism of a determined martial artist and warrior, someone who drove herself to be the absolute best. It showed vanity and an uncompromising nature, but also determination and relentless self-discipline. I don't think it showed her going into a rage just because of a single hair. But I'll agree it showed she was tightly wound. Someone with such unrelenting standards of perfection would plausibly have trouble dealing with it when things started to unravel.
Then later on Ember island she was clearly not playing with a full deck of cards.
I'll grant that. I didn't care for her portrayal there because it seemed out of character to me; I saw her as this very smart, canny, rational character, and it seemed to diminish her to have her act like she did on Ember Island. But thinking about it in context now, I suppose it was more a case of foreshadowing, exposing the instability that lay beneath her tightly controlled surface. I guess it's when she's outside her safety zone that her poise slips. Maybe she's sort of a hothouse flower, in an odd sense -- she's always lived within this safe cushion, surrounded by people who gave her everything she wanted, and so she never learned how to function well outside of that. Sure, she managed pretty well following the gAang with just Mai and Ty Lee along with her, but they both obeyed her wishes and kept that same sort of comfort zone around her.
Still, it disappoints me to have to redefine Azula that way in my mind. I was so impressed by her as a truly smart, capable, rational villain without the usual self-defeating villain hangups or Evil Overlord tendencies. And I kind of had a crush on her too, since ultra-competent women excite me. (What? I thought she was older. Plus, she's a cartoon.)
Of course as far as a general compression of events goes, it may be worth remembering the ENTIRE SHOW takes place in less than 12 months. Now that's pushing things.
Yep. The winter solstice is a third of the way into the first season, and Sozin's Comet comes at "summer's end," which would be the autumnal equinox. So assuming their years are the same as ours, that's 9 months plus however long the first 6 episodes took.