When Alex Rosewater tries to start up Big Fau, Big Fau goes through the whole "Cast in the name of God" stuff before going into "Ye Not" before shutting down. It's not the fact that Big Fau shuts down automatically due to missing components, but is actively rejecting Alex with the "Ye Not." An executioner's weapon would be blessed by a priest, so when it was used, the executioner's role was ordained by God, and thus, not a murderer. Hence "Ye Not Guilty." Alex thinks he IS a god and thus the practice falls apart.
Alan Gabriel tried to use the the instrument of God for murder. He was thus found guilty of this sin ("Ye guilty" in contrast to the usual "Ye not guilty") and was himself slain by the executioner's weapon he tried to abuse.
Roger's Catch Phrases "It's Show Time" and "Action" make a lot more sense at the end of the series.
In episode 11, when the sax player and his blind girlfriend are eating, there's no food on his plate, but he makes motions and noises with his utensils as though he's eating. At first I thought this was an oversight by the animators, but then I realized: they're dirt-poor; he's secretly giving her his food.
The Big archetype is extremely fast and flexible but the Bigs themselves aren't. I can't think why it would be so flexible but the speed makes sense since while Bigs are slow but they have heavy weaponry and armor to slow them down.
The Archetype might have been a Super Prototype, with the Bigs being (relatively) dialed back Production Models. Not as much raw power, but you can afford to build an army of them. It could also be that the Archetype was the base chassis for the Big series. Its increased speed and strength comes simply from the fact that it isn't lugging around all of the weight that gets bolted onto the O, Fau, and Duo classes. All of them, especially the O class, carry extreme amounts of armor. That would necessitate an extreme amount of power on the part of the mecha to ensure reasonable mobility. The Bigs were military vehicles afterall.
The first time Roger meets Seabach, the later announces his name is "Schwarzwald." Roger looks up and says, "black forest?" which is what that means in German. So... there are foreign languages in a city where they're unsure if outsiders exist? It only gets worse when Dan Datsun identifies Michael Seabach of being if German descent, which means on some subliminal level foreigners make sense even though officially no one is sure.
It's possible that there are simply people who are of German decent who were living in the city when the event happened, like the Japanese in the joke episode. They might have forgotten about Germany or at least do not believe Germany still exists, but are simply aware that there are German people. Insert Hentalia reference here.
No one is sure that there are outsiders anymore.
Indeed. They knew that there used to be outsiders, but much like Romdeau in Ergo Proxy, the status of the world outside the domes is somewhat up in the air. People in Paradigm seem to default to the idea that they were the only survivors.