If Sozin had killed all the airbenders, and then worked through the next three Avatars, would the cycle have stopped? Skipped over? Thrown a Null Avatar Exception?
In canon it is stated that the only way to end the Avatar Cycle would be to kill the Avatar while they were in the Avatar State. A specific mention to this vulnerability is by Roku during "The Avatar State", stating it is a time where the Avatar is most vulnerable. (Comment by Micah).
As in-universe, the Avatar is a literal manifestation of the Earth's spirit in human form, they may only be totally destroyed when the whole of the Earth's spirit is present, the Avatar State.
However it is also mentioned in canon, that if Ozai were to destroy the Earth Kingdom, it would totally and irrevocably alter/destroy the balance of the universe they exist in. So it is entirely possible that this balance was only kept alive due to Aang's survival during Sozin's attack on the Air Nomads. So it is not entirely possible that killing off all the airbenders would not only break the Avatar cycle, but also possibly the entire balance of the elements and cause unknowable effects on the environment they exist in.
Edit: In light of Legend of Korra's revelations in Book 2: Spirits. It looks like the destruction of the Earth Kingdom would upset the balance between the elements and would prevent Raava from being able to regenerate into another Avatar. Also, once bending is granted, it can be passed on through genetics, possibly due to a piece of the spirit/chi passing onto the next generation. So in this sense, they can't get back Airbending other than Tenzin's genes and that is all the Airbenders were killed, that the current Avatar Cycle would definitely end, though if Raava was able to find a suitable other vessel (unlikely since only the Energybending Lionturtle is alive to give bending), a new cycle could begin.
I believe it could simply skip to the next element.
There doesn't seem to be a reason for the Avatar cycle to work the way it does. After all, the Avatar is basically the combined spirit of Raava and all of the previous Avatars, flitting from life to life. I don't see any reason why the spirit couldn't be reincarnated by skipping an element.
Besides, there has been no concrete indication that genocide can wipe out a bending ability permanently (Attempts by Sozin and Amon seem to indicate that this is true, however). The lionturtle(s) certainly can intervene and reintroduce the ability (not so unlikely, the mass genocide would lead to a great imbalance which they probably do not like). The current Avatar may be able to use energybending to do the same, if alive. Finally, benders may "reappear" through a recessive gene, though as far as I can tell the bending ability doesn't seem to work that way and only appears if one of the parents has it.
Also, I think that Aang was born the precise moment Roku died, so killing all the airbenders would have just killed Aang and moved the cycle over to the waterbenders. Not sure of this, though, and it's not so relevant.
This answer is in addition to the well thought-out and detailed answer by ardentsonata.
Killing off all the Airbenders would NOT "break" the Avatar cycle, but MIGHT have led to a situation where the Avatar could not REINCARNATE. It's basically the difference between blowing up a plane and keeping it from landing. The plane may not arrive either way, but it's important to recognize that the two ARE different things.
Perhaps part of the original plan was to hasten the return of a Fire Nation Avatar, but Sozen's TRUE goal may not have been so simple. Remember, Aang was 12 years old when Sozen BEGAN the genocide of the Air Nomads - meaning Sozen had already waited 12 YEARS to put this plan into action:
Element bending can be given and taken away through Spirit bending.
Avatar Aang was seemingly the first human to be given Spirit bending by a Lion turtle, though Amon seemed to be able to sever a person's -physical- connection to an element via blood-bending. (Possibly via some form of lobotomy)
Amon was able to Physically remove Korra's bending but Korra was still able to form a NEW connection to Air.
The Avatar reincarnates through all 4 nations. Legend of Korra Book 2 expands this to (seemingly) include ALL humans.
The Avatar ALWAYS masters the element of the nation they are born into first.
A - This implies that bending is somehow carried from reincarnation to reincarnation and is affected by both the Spirit and the Body of the bender.
(i.e. bending ability and the NUMBER of elements is a product of the Spirit and persists from reincarnation to reincarnation but precisely which element a bender can bend is a product of the body aka. genetics)
Please also consider:
All humans reincarnate upon death.
at the end of the Hundred Year War, the Earth and Fire Nations had a HUGE population
The Southern Water Tribe had a very small population, and few water benders.
The Air Nomads had been exterminated to the point where AT MOST a mere handful survived in hiding amongst the other nations and not a single Airbender had been seen in decades.
B - IF all humans reincarnate through the Avatar Cycle, all of the above is explained. The Air nomads die and are reincarnated into the Water tribe. When Water and Earth go to war against Fire, Water casualties become Earth warriors and when Earth warriors die, they become Fire Nation. Yet Fire Nation casualties do not reincarnate at all, leading to a sharp drop in Water Nation birth rates. Ultimately, the Fire nation steps up to a Total War footing and mobilizes their entire available population in an endless stalemate against the massively swollen Earth Kingdom population.
C- Killing the Air nomads may have been part of a long-term strategy of attrition warfare on the part of Fire Lord Sozen. In trying to eradicate the next Nation in the reincarnation line, Sozen guaranteed that the souls of his own warriors would not be reborn to fight against him. Meanwhile, his STANDING army only stood to GROW in size.
D- If Phoneix King Ozai had been successful, the mass murder of Earth Kingdom citizens would lead to a temporary baby boom in the Fire Nation, but would greatly hasten the eventual destruction of the entire human race.
E - When you call Tenzen's wife the "Mother of a Nation" try two. Literally.
I don't think that it would cause there to not be another airbender Avatar. It IS possible to learn an element and be a bender in that without having been taught, as you can see from Katara in ATLA, and again by Korra in LOK. As a child, Korra was earth bending and firebending, and that is how they knew she was the Avatar. So in theory, there could be another avatar, who would have to learn airbending on his/her own, but could possibly never be a master.