Wow, and I thought I was bad for skipping words while typing...
One, Yes way they would be the same, they share the same original manufacturers, do you THINK that they're making laser rifles in Fallout? No they're finding them in military institutions and whatnot, they don't have the ability to make much more than a finely crafted .223 special even after 50 years of re-learning!
Two, the 7 year old girl bit if I am not mistaken is that how would a 7 year old survive the trek, surrounded by xenophobic men in power armor, how would she survive the radiation, the diseases, the lack of food and water, not to mention the regular raider attacks, as BoS stick out like a sore thumb in the desert, hello chrome on sand!
Yes USA has changed it's structure (I'm assuming that's what you mean) several times now, however this is a technophile religion we're talking about, not an institute of public governance (and the occasional scandal), how often have you seen the Roman Catholic Church change it's initiation or methods before the turn of the century?
It's not undecided, they got two options, to foot it or to Brahmn and cart it, however both options it is probably only a 5% success rate of reaching there, and probably under 1% of the entire group making it there in one piece. Vertibirds are out because of their massive fuel consumption (do you realize how much gas it takes to keep a normal helicopter in the sky, let alone something like a Vertibird?)
The Chinese (perhaps...) Super Mutants, how the hell would they cross the ocean, last I checked the Chinese were not in the process of invading America before the bombs fell, instead they were covertly creating their own muties after stealing the tech from the American Scientists(I am not 100% on this one mind you, I think I read it in one of Beth's excuses for the Washington Muties, but I am likely mistaken).
How did those pesky muties cross the ocean, in all likelihood they'd be too busy rampaging throughout China rather than hunkering down in Washington, perhaps a small contingent like chinatown in San Fran could have landed, but there's no way an entire army would have breached shore WHILE the bombs were dropping...