Here are my answers to your questions! If you have more, just ask, I'll be delighted to answer!
★Unlike the canon timeline, Aang is conceived sometime before Roku's eightieth birthday, so he's now only about two years older than Azulon.
And yes, just like Yue in canon timeline, Aang is stillborn. Perhaps the universe senses the imbalance wrought on by Roku's miraculous survival and tries to correct its mishap in this cruel way—a warning, a punishment.
However, it doesn't last long.
Yet, unlike Yue's parents, the Air Nomads don't have two Great Spirits turned koi fish swimming in a pond to grant a new life.
Despite that, one has to remember that Air Nomads are highly spiritual people and every(?) child born to them is an Air Bender.
Perhaps Aang's parents pray really hard—or at least enough—that some Great Spirit of the winds answered their pleas with a miracle.
Perhaps Raava finds a way to help the boy that could've been (should've been) hers, pushing a spark of her Light into his all too-still chest, to restart his heart.
There are many possibilities and many unanswered questions.
Roku once catches Wan—the jovial swindler full of kindness and snide remarks—stand over Aang's sleeping body in his glowy spirit form. His eyes are wide and ablaze with such violent hatred, it makes Roku's blood freeze in his veins.
For one reason or another, Wan wants Aang dead.
★I've already mentioned that during Gyatso's travel to the Ember Island, he had to settle down and sleep inside inns for the night—he had coin to pay, so it was alright.
Keep in mind, Sozin had been keeping an eye on Gyatso the moment he set a foot inside his country.
He is well-aware he [Gyatso] used to be a part of Rokrew and was one of Roku's closest friends, as well as Ta Min's pen pal(?).
He suspects Gyatso was in the Fire Nation to inform Ta Min that the Air Nomads finally found their Avatar.
Maybe he [Sozin] hopes that Gyatso carried some form of a letter or a document that could determine from which Temple the child hailed from.
But what his spies bring him instead was... to put it simply... so much better.
(I tried to be a little bit vague-ish, to keep some semblance of mystery.)
★I think that before I can attempt to answer this monster of a question, I have to make a very important point—which is more likely one person's opinion—still:
I don't think Ozai believed in Sozin's imperialistic ideology with his whole heart and mind (unlike Sozin, Azulon and Ozai's children) and simply used his grandfather's propaganda to further solidify his own power and claim to the throne. Ozai used imperialism as a tool to further his own success. I'm sure he was taught to believe in the Fire Nation's eternal reign, considering he was raised in such a system, but that belief was eclipsed by his lust for power, glory and control. And his own image, of course.
Which also leads to another very important distinction between Sozin and Ozai that I believe is very important for this AU:
Sozin, unlike Ozai, isn't wasteful when it comes to the Great Comet's firepower.
In canon timeline, Sozin planned for his grand moment for twelve years—mapping out the battlefields (the Temples) and their exact locations, learning as much as he could about the enemy (Air Nomads) and their culture, so he knew how and when to strike. He studied the Great Comet, the records lf its last appearences, the effects of its power.
Sozin wasn't wasteful when he comitted the genocide. He knew exactly what he was doing, targeting the Temples as he did.
Ozai on the other hand? He wanted to feel the power, the ecstasy of being in total comtrol as he rained pointless terror all over the Earth Kingdom, showing off to his enemies just how powerful he was.
His invasion was a pointless excercise of violence and I don't think (again, simply my own opinion) that it was planned very well.
He wasted the Comet's power to burn it all to the ground when he could focus on destroying the most significant places (from cultural and military standpoint), like Ba Sing Se or Omashu, instead.
As I've already said in my last ask's final point, Roku received a letter with Imperial seal stamped upon it.
The context of this letter will be kept a mystery for know, but I can at least tell you that Roku was incredibly unhappy after reading it.
Roku and Sozin reunite after twelve long years during the night of the Comet.
Sozin, mounted on his blue dragon (whom I like to call Talon, 'cause it's a cute headcanon, I saw others do it too), arrived with a small army of firebenders backing him up, some of them riding dragons too, ready to retrieve their "abducted" Avatar to the safety of the Fire Nation.
Roku finds out only a little bit too late that Sozin's "attack" on the Southern Air Temple was only a distraction, serving for a grander purpouse.
The Air Nomads may have survived that night, for ever changing the course of history, but another nation paid the price.
★For this AU specifically, I have a headcanon thay Azulon's mother died in childbirth—which sort of leads to the popular headcanon for the canon timeline where Fire Lady Ilah died giving birth to Ozai, and that's why Azulon resents him so.
Due to the intense focus on development of the military, science; industrial revolution in A:TLA world, the drill, the air balloons and air ships ect. I believe that some branches like medicin and healthcare were pushed aside in favour of growing a stronger army.
So, at that time, it was rather common—even amongst nobility and royalty—for women to die after intense labour, especially if they weren't given proper medical assistence.
I believe that since the Fire Nation is heavily based on Imperial Japan, and also borrows influences of the likes of China and Korea(?), I can safely place here the headcanon of imperial harem within the royal family, so yeah: Sozin has concubines and consorts, and with few of them managed to produce some children throughout his life—Azulon is the youngest, and the only child he had with his Fire Lady.
(Two concubines and one consort had the misfortune to die in childbed as well.)
Which, is somewhat a stroke of luck, conskdering Roku will be crowned Fire Lady by the Fire Sages shortly after their wedding ceremony.
Sozin would've loathed to arrange an accident for his previous wife; her clan is very influential, so he couldn't have demoted her to a concubine.
Her death in childbed was a mercy in his eyes.
The hierarchy of the harem itself remains largely unchanged, and the alliances with the more powerful clans still hold—these concubines and consorts were naturally selected more for their name and its power/influence wituin the country rather than physical beauty—so the delicate structure hasn't been largely damaged.
However, the concubines and consorts themselves are displeased by this new union, which just happend to be with THE Avatar, nontheless—who, by the way, everyone thought was dead until now.
They fear they're starting to lose Sozin's favour and whatever affection he'd held for them (which is true, he has declined to share their company for some time now.) They know they are being replaced and they don't like it.
However, they believe that they can gain favour of someone far more powerful if they just try hard enough.
Just how hard can it be, hm? Seducing the Avatar.
★Yeah, Roku's family are incredibly unhappy with the news—Oooh, Ta Min is positively furious—but they know that as long as Sozin has Roku in his clutches, there's nothing they can do alone.
Well, Ta Min thinks, now's the best time to catch up with some old friends.
Again, sorry for my grammar and spelling mistakes, I originally wrote this at 2 AM and fell asleep, only sending it just about now. :<
No problem with the errors