Maybe they wanted the launch codes so desperately so that they could stop the war from happening. (theory)
The best way to stop the activation of our nuclear arsenal is to destroy the arsenal itself. If that is not feasible, the second best solution would be to destroy our command missiles (these thingies have payloads that can authorize nuclear retaliation, nasty) and then kill the people would actually fire the missiles. (those launch codes are worthless if no one can turn the keys and fire the missile)
I think the most important question is how they're related to the Great War. They've obviously know about Earth from way before we even had nukes.
Did they start the Great War by getting the U.S. launch codes and fire on China or were they trying to prevent us from blowing up Earth 'cause they wanted it or wanted humans. A part of that question is also answered by what the purpose of the Abominations are. Since they're also hostile to normal aliens on MZ, my guess is that they're to be used as shock troopers in a land assault on Earth, I don't think they're workers nor the next step of alien evolution.
Perhaps they "just" want Earth's resources, but were too late and we blew ourselves up much to their dismay, but they stick around since they can't go back, perhaps their world is even worse off, also due to world wide mass destruction. I'd call that irony and it'd actually fit into the Fallout universe in my opinion. Perhaps the destructive nature of alien couldn't sustain their society either.
The abominations might also be hostile because their absolutely insane. Audio-logs imply their created by the introduction of a mutagen into the specimen, who then undergoes some sort of horrible transformation. Insanity is a logical result considering the immense stress this would trigger, as well as the effects of having the brain, bodily chemicals, and other factors effecting behavior suddenly change on someone.
As for the great war, I'm going to guess their actually not related. The nuclear launch codes aren't some magic key that activates an automated launch program. Ultimately, the decision to fire the U.S Nuclear arsenal is up to two guys sitting a bunker somewhere relatively near the weapons they can launch. My guess is that they have some sort of long-term scientific mission to satisfy, and our technology is relevant to their mission. Misinterpreting the robo-ponies as weapons could explain that weird room, and they had large stockpiles of our tech. Also, these guys have been orbiting earth since at least the era of feudal japan. Why would they suddenly nuke a population they've been studying for at least a few centuries? Or invade us? The abominations, their interest in our tech, and the whole collecting people stuff makes me think their xenobiologists.
A long term deep-space exploration mission consisting of at least two huge ships with a complement of minor craft and large crews couldn't justify itself simply by collecting data and specimens. The abominations must be part of an effort to discover practical applications for the data they've accumulated on earth. After all, creating something that is much stronger than your own species can have a wide range of practical uses. The reason why they made the abominations, store people in cryogenics, make them do interviews, and all sorts of other experiments is because they want to have something presentable when they go back.
I really don't think their interested in invading earth. Attacking a foreign planet is one of the most hugely inefficient waste of resources I can think of. Any planet that evolved it's own sapient life is most likely completely self-sufficient, that gives the natives a huge advantage over any invaders, who are light years away from reinforcements and production facilities. As for wanting our resources, that's far too impractical. traveling at the speed of light, they, at the very least, have been traveling for 4.3 years before reaching earth. Most likely longer (considering their probably not from Proxima Centauri). To satisfy a planets needs would take a fleet of ships, and we've only seen two and have no reason to believe any more are in proximity to Sol. Taking into account the trip back, then if they were here for resources it means a minimum of 8.6 years travel time. and again, the travel time is most likely a lot longer. That 8.6 years doesn't account for all the time that would be spent gathering resources and loading them into ships. Finally, you need to consider the fact that earth is simply not a practical target if your looking for resources. Your own solar system would be the first target, as it's going to be made of similar things to your native planet. Also, unless their from some really nearby star, there are probably closer planets.