You're asking questions that the game doesn't provide answers to, unfortunately.
As much as I would love to know how Mobius got their start in Cedar Hill, too, the game straight up doesn't even touch on it. They don't even come close
to touching on it. The name "Cedar Hill" in and of itself only appears in two newspaper articles; otherwise there's no mention of the name of the church at all. It bears mentioning that Cedar Hill was never explicitly stated to be in Elk River, either, but it's the only logical conclusion that we can draw, considering what Kidman tells us about her past and her hometown and the church. Details around the Elk River/Cedar Hill cult are very, very spotty.
But I suppose we can try to tease this out a little bit. Bear with me, this post is about to get LONG.
My only guess is that Elk River/Cedar Hill is where Mobius more or less got started/planted its roots. It started as a cult and morphed into something much bigger. Based on all of the information we're given (which isn't much), that's the best conclusion I can come to. And this conclusion comes from Jimenez saying that research at Mobius dates back to well over a century + Joseph's observation that the church and the graveyard seem straight from the 16th century. As we know from general world history, there were very, very few European settlers in North America in the 16th century (and from the way the devs talk, we know that TEW takes place either in the US or Canada); most settlements began closer to the 17th century. So, based on Joseph's observation, we can probably safely assume that Mobius started as a European organization and came over to North America sometime later.
So, let's move forward from there. The only other link we have to Mobius being a global corporation, other than the mention of the 16th century, is a newspaper clipping that you find closer to the end of the game about a building burning down that was rumored to be a front for foreign intelligence. For reference, here's the transcript in its entirety:
FIRE AT OFFICE
OFFICE LINKED TO FOREIGN INTELLIGENCE BURNED
Building housing community outreach center burns to the ground. There were allegations that the center was a front for foreign intelligence.
What's perplexing about this newspaper clipping is that I can't directly link it to anything else. There's no indication of where this office was (whether it was Elk River or Krimson City) or when this happened. This incident is never mentioned or even so much as alluded to outside of this one newspaper clipping, but considering the fact that Mobius is always, always linked to buildings burning down, we can assume that this place has something to do with them. It's just more evidence that Mobius has ties or interest in Europe and possibly even started there.
So, what else do we know?
We also know, thanks to Jimenez, that Mobius has other branches of study outside of STEM. I remember him specifically citing natural/botanical studies and things like that. But they all seem to be linked back to the purpose of mind control, which is how memories 400+ years old could end up in the modern invention of STEM. With regards to how the metaphysics of that all works, don't ask me; I don't understand the metaphysics of STEM at all. I'm not a science person, but even if I was, Mikami's special brand of pseudoscience is kind of a mystery in and of itself anyway. (Remember when the Progenitor virus in RE was actually manufactured from Ebola? And then became ??????????????????????????????????? IDK FLAME SPITTING BLOOD VERONICA VIRUS)
But back to Cedar Hill. Cedar Hill may have been some sort of offshoot of Christianity, since its religious leaders are given the title of Pastor... but Christian churches name themselves as such explicitly. The question then becomes: was Cedar Hill originally a Christian church before it became Cedar Hill? Or was it something poorly masquerading itself as Christianity but never ever actually was? Was it just Mobius the whole time? Did they come to North America seeking "religious freedom" even though they were an evil fuckin cult? I actually think that might be the case.
The whole thing seems very Silent Hill to me. European settlers come to North America and base a community around some fucked up church cult. It all goes well for hundreds of years, until it suddenly doesn't anymore. But by the time the church floor collapses, Mobius at least has enough money, a strong enough base, and a far enough reach to not completely collapse and royally screw themselves the way The Order in Silent Hill does. So they move onto Beacon and other places as well. But since I can't objectively prove that one way or the other, I didn't put it in the essay.
Now, let's be clear: The leader of Mobius has no supernatural powers, and this has nothing to do with a God. There is nothing supernatural in The Evil Within; it's science fiction, not fantasy magic. For most of the Kidman DLCs, she's not even interacting with the real Administrator. Ruvik tips his hand very late in The Consequence to show us that it's actually him. "The Administrator" says that the injection that Mobius gave her wasn't to protect her from Ruvik's influence; it was so that she could bring him (The Administrator) into STEM with her. But that's not true. Ruvik tells us in the second Torn Letter that The Administrator actually entered STEM himself seeking some sort of power "long ago" and that Ruvik left his mark on him and now has a vessel in him. Now, I've seen theories that Ruvik is some kind of a psychic in the same vein as Alessa Gillespie, but there's absolutely no basis for it in canon. Everything in the Kidman DLCs except for the very final scene of The Consequence takes place in STEM. It's not supernatural. It's just STEM.
There's no hole in the essay with regards to the construction of STEM. I say at the very top of page 4 that STEM as a project began in the Victoriano mansion basement while Ruvik's parents were both still alive. Going by the ages in the model viewer, Ruvik was 18 when he killed his parents. Ruvik himself wasn't killed until he was 37. That's at least a 19 year development cycle for STEM, and Jimenez's research with Ruvik's data continued for what I would assume to be at least a few years after his death. Mobius didn't modify anything for STEM with regards to the wireless system. Ruvik developed that on his own, then Jimenez stole the blueprints after he was killed and built it while keeping it a secret from Mobius himself. But, of course, they eventually found out, and that's why Jimenez set it off.
Sebastian was pulled in wirelessly, yes. Jimenez explains early in The Consequence that a person gets pulled into STEM wirelessly via use of a specific sound frequency, and Kidman assumes that that's what the loud screeching feedback noise from the opening cutscene of the main game was. From one of my conversations with Rikitatsu, I came to the conclusion that Mobius had to turn off the wireless STEM sound frequency in order to enter Beacon themselves, lest they all get sucked into the system. But they still saw this as on opportunity not only for STEM research, but for Sebastian to oust Ruvik from the system and maybe even take over as host -- and so they hooked everyone who was still alive into the wired one. There's a whole bunch of monitoring systems attached to each of the STEM terminals, giving them access to psychological and biological data during the game's progression, essentially.