1) The Discovery and it's mission is connected in some way to Section 31 (the ship's registry number is NCC-1031. Coincidence? Hmm).
Note: the following may contain
minor spoilers if you haven't seen episode 3 of Discovery, but nothing too Earth shattering.
I agree completely with this, in fact I think it may even be a Section 31 ship. There are actually many clues to this. Lorca tells Buhrman to "not worry about Starfleet command" when he's trying to get her to join the crew. That would make sense if he's working without the knowledge of Starfleet Command. Also, some of their research is illegal, not a concern to Section 31. When she's walking through the ship, Buhrman sees people with black Starfleet badges, and the camera focuses on that a moment. Black uniforms and signage are often associated with Section 31.
Now, I don't believe the entire crew is aware of Section 31 or why they're there. I think most have been fed a lie that they're working for Starfleet Intelligence on top secret research to win the war. But, I think there's more to it than that. I believe Lorca definitely is either a member of Section 31, or is at least aware of its existence and is taking orders from them. His security chief Landry may be too, she seems "in the loop." I don't think Saru would have knowledge of them, as that doesn't seem the kind of thing he would be into. Though, I may be misreading him. Most certainly, the whiny science officer Stamets isn't a member of Section 31 and I see no reason the young Cadet Tilly would be involved with them. She's likely on there because of her scientific qualifications.
Anyway, that's my theory. We'll see how it goes.