Ghost in the Shell: Innocence uses some brilliantly subtle visual cues easy to miss if you don't pay close attention. For example, though it's not relevant to the plot, you can tell that Dr. Haraway is a cyborg at a single glance, since she's sitting in a cold room where Togusa has to wrap his coat tight and his breathing becomes visible while wearing a labcoat with sleeves rolled up and top button open from her shirt without signs of discomfort - and her breathing, as well as Batou's, is not visible.
Episode nine of the first season of Ghost in the Shell: Stand Alone Complex is an entire half hour of Fridge Brilliance. For one thing, although it becomes obvious by the end of the episode, it's never made clear that the whole thing took place on an internet chat room. The viewer simply infers it after a few minutes, just one must infer that the pink-haired woman is Major Kusanagi due to the fact that she obviously shares the Major's voice and mannerisms. But it goes a lot deeper than that. The voice of the fellow who seems to be overseeing the discussion does not go with his appearance at all, and the first time you watch the series, you might assume it was due to a fluke in the dub (surprising for a series with such an amazing dub). It's actually the same voice of the boy who Togusa meets at the beginning of episode eleven, who turns out to be a friend of Aoi, the (true) Laughing Man. As for the Laughing Man himself? He's closer than you'd think. The stout, middle-aged man should strike most discerning viewers as somewhat suspicious due to all the information he seems to have, but he never appears again in the story... apparently. Remember that online, a person's identity can appear completely different from what it is in real life, except their voice. The online personae of the Major and the boy from episode eleven retain their real voices, and the kicker is that the middle-aged man in question is voiced by Steve Blum. So is Aoi. Although it's never made clear, it seems very likely that the Laughing Man himself was listening in on, and even participating in the discussion debating his true nature by his... fans, for want of a better term. It's understandable, since he's so curious about the stand alone complex he had inadvertently created.
The intro to the 1995 animated film features the Major's body being manufactured in a factory. Or at least, we're lead to assume that is what we are seeing. Much of the film's themes center on the Major's sense of self and humanity, and one scene prominently features her encountering a doppelganger of herself in passing. It's entirely possible that the intro didn't feature her at all, establishing the film's theme of human vs machine.
On the page for the original film, the Fantastic Nuke listing talks about the fact that computer programmers are as tightly restricted as nuclear weaponry. This is easily understandable when someone like the Puppetmaster could take over people's brains. None of the people it takes over are important but when you think about it, it's both Fridge Brilliance and Fridge Horror when you realize they could just as easily hack politicians, government agents, generals, admirals, anyone who would have power to use real nukes...
If the Major is a cyborg with a machine body and a human brain, how can she have periods?
They are a reoccurring mental pattern? I mean, she can't really have them due to her artificial body but her brain remembers having those and it might be calming in a paradoxical way.
Alternately, she doesn't have periods, and her claim of being on her period in the first film was just some dry humor.
In most (all?) of the continuities, the Major has been a full cyborg since before puberty, so she wouldn't remember having periods because she never experienced them. On the other hand, unlike many of the other full or partial cyborgs, aside from the dataport on her neck the Major's body is indistinguishable from a normal baseline human. If she wasn't joking, it's possible that her body is programmed to intentionally replicate having a period so she can feel more human.