Red squad didn't exactly have "their own starship" officially. They were originally on a training cruise, and they commandeered the ship after all the officers aboard died, instead of letting Starfleet know and getting further instructions (which probably would have told them to come home). Kor
Leyton was most likely playing the long game. He must have been working on the conspiracy for years. I find it VERY likely that Red Squad was created specifically as part of the plan. It's entirely within Leyton's wheelhouse. The reason I feel thsi way is that Starfleet, as a whole, doesn't seem like the type of organization which would ever create something like Red Squad. Seems rather elitist. As for the Valiant? Others in the conspiracy could continue Leyton's work. So even if he personally couldn't have assigned the Valiant, then others must have.
Interesting. Perhaps the 2372 coup was Leyton's Beer Hall Putsch, and he had plans to return with a vengeance (although, there's no evidence he has done so by 2399). Could there be several Leytonists (for lack of a better term) in high positions of power fulfilling his agenda? The Star Trek Litverse has Leyton retiring into obscurity following prison, but there were questions regarding his exact whereabouts and future plans.
Not exactly questions, everyone knew where Leyton was. Indeed, it's a plot point that he's frequently hounded by reporters, journalists and historians wanting to interview him, which he ignores. Indeed, when Section 31 files were leaked to the public in the litverse, Leyton ironically ended up being one of the cleanest Admirals from recent times in that he had absolutely no connections with Section 31 and attempted coup aside, was not corrupt at all. Indeed, the reason Leyton got as far as he did with the coup was because Section 31 was too busy dealing with other Starfleet Admirals who were red-flagged as greater threats. Also, according to the novels, the coup was motivated by Leyton's genuine belief that the Federation wasn't ready do deal with the Dominion. With that in mind, he would not have been able to be planning it out any longer than a year and a half, the amount of time the Federation was aware of the Dominion prior to the episode. And probably less than that, since he presumably didn't start thinking about a coup until after Jaresh-Inyo began rejecting his recommendations regarding the Dominion.
FWIW, when i was in the US Army we'd sometimes get West Pointers attached to our unit during the summer (between their third and fourth years, IIRC). They were essentially treated as "third lieutenants" for the summer and would spend their time shadowing our assigned officers and getting a sense of the routine in an active duty unit. If this was ever done during an actual/real-world deployment (i.e. combat) I've never heard of it.
I haven't actually read Available Light (I'm finding myself having a hard time investing both financially and emotionally in the last few Litverse novels). Memory Beta, for some reason, implies that Louvois said that Leyton's "whereabouts were unknown", although that could just be a reference to his obscurity and hiding from nosy reporters. The coup was both well-planned and very hasty, which makes me think he was putting pieces in place much longer than 2370 or so. Maybe he didn't think the Federation of the 2360s could face another Borg invasion, or even was appalled at the events of 2364 with the Bluegills. He definitely seems to be all about good intentions, so Section 31 probably wasn't interested in recruiting him, but maybe manipulating him.
Always nice to get some perspective on how these things work in the real military. Thanks, @Mysterion!