We find out that STEM wanted a human body. That's all well and good, but what does it plan to DO? Just live as a human? Take over society with its hacking skills? Maybe this was covered in the movie, and I missed it.
"Just live as a human?" Yes.
Even if STEM wanted a pure human body for some reason (this was explained, but I forgot), why go after Grey? Why not have Eron Keen look for a more willing participant? With Vessel's money, surely they could find someone willing to give up control of their body (paraplegic or not) without all the games STEM had to pull with Grey.
Because STEM wanted complete control over a human body and to live as a human. As Grey does to some limited degree towards the end of the film, even if STEM controls the majority of the body, the person in question can resist long enough for outside factors to interfere by, say, knocking him out, killing him, or somehow destroying STEM itself without killing the person its bonded to. And STEM is smart enough to know that its host, a willing participant or not, would effectively never surrender their very personality and let STEM have complete control if said host started suspecting STEMs true intentions. For STEM, its a lot easier to work from angles where humans can make very irrational decisions, such as revenge, and play up being the helpful A.I. who turns said host into a cybernetic superhuman. As for why Grey specifically, its likely that Grey and his wife were the only two people, out of a very select few, that STEM knew who were not suspicious of STEMs true intentions to begin with, AND could leave the compound without much interference.
STEM's number of potential patients is very limited. As he mentions when Grey tells him about the Old Bones bar, STEM can only access people with an online presence. Since most people as backwards as Grey would avoid social media and websites, he makes a good subject. A vintage car restorer is someone who fits the cross section unlikely to want implants and also can be found on the web.
There was an implication that the police drones can actively track people through their various cybernetic implants. There are ways around it, but STEM was looking for an otherwise unmodified body in part for that reason, digital anonymity.
Not just an implication - Cortez outright says that everyone has an ID chip in a tooth filling. That's why the assassins used those masks, they blocked the signal from the chips.
Why the hacker?
If STEM was running the show the entire time, then why did it need the Input Guards removed (which is what the hacker did)? Surely it wouldn't have been letting Keen put those in in the first place. Even if they were put in before STEM got super-intelligent, why not get Keen to remove them?
Keen probably put them in place as part of STEM's creation and never removed them to insure it never turned against him.
Keen was super upset when STEM killed those guys. As far as he was concerned up to that point, the experiment was going along harmlessly. Furthermore, if STEM had asked for them to be removed beforehand, it would have aroused suspicion in Keen, who was shown at the end willing to kill to end the experiment, and cut the whole plan short.
Arm Gun Ergonomics
What exactly is the point behind a gun built into your arm? It can't be concealed without gloves and a long sleeve shirt (the latter of which would cover the loading port, and therefore defeat the purpose of such a weapon), and the fact that reloads have to be conducted bullet by bullet puts it at a severe disadvantage against a modern pistol or rifle. About the only advantage such a weapon seems to have is that it can't be dropped.
The arm guns don't have to be draw, don't take up any space on your vest or legs, and seem to carry plenty of ammo between reloads. The concealment was good enough that no one picked up on them being armed at the crash site as well. Other than that, they were said to be experimental implants. It's possible that the augments weren't considered feasible or were in the process of being improved.
Being armed while still appearing unarmed from a distance would be a major advantage for any attacker looking to ambush their victim or keep a low profile.
There's another problem to it, which is that it's much harder to aim an extended arm accurately than it is to aim a rifle. Granted, they only seem to use pistol and some sort of shotguns rounds for them, and there's the inevitable "cyborg accuracy" handwave, but still...