I think that's what the reminders about River were about. It took me awhile to make the connection -- I couldn't understand what the framing Missy story had to do with the simulation story, why thinking about what was said at the execution would help sim-Doctor figure out to send himself an email via the glasses. He pulled out River's diary, quoted it, and emailed himself from inside a simulation. How could he do that? The simulation was "too good", as he said, and dealing with Time Lord tech can have unforeseen consequences, where even the simulated version can have real capabilities. Just like simulated River, in her after-death virtual state. She could still reach out to the real world and communicate with him... so he thought just maybe he could do the same thing, and communicate with himself.
It was very clearly answered, though it's more clear on re-watch. At the start of the episode, he gets the email in his glasses. There's a pixel-y dissolve to the credits. After the credits, it's a pixel-y dissolve again, to the scene where he's at the lecture podium and the pope comes in. This is what he is "watching" in the email he just received. The rest of the episode is what he is 'watching' from his glasses, basically we're just watching it with him -- until the end of course, when we go back to reality.
He said it was a psychic recording, which is - presumably - why we could see stuff that the Doctor himself didn't... like when sim-Bill and sim-Nardole were at sim-Cern...
Yes we know why he made the oath. He swore to protect the body for a thousand years (whether or not that 'body' was actually 'dead'). He used the very words "I give my oath" or "I make this oath" or whatever. That's the oath.
My own question is: why would these bad guys, in making a simulation of earth/humanity as real and accurate as possible, include a book that tells the sim-humans that they're not real?