When Walker saves Ethan in the fight at White Widow’s palace, no mention is given as to how he managed to get past the security without causing a scene outside. On rewatch, knowing that he’s Lark reveals that he was allowed inside without issues.
On the plane before the drop into Paris, Ethan mentions that the only reason why various intelligence agencies haven't found Lark yet is because Walker accidentally killed an informant that was going to rat out Lark. In the moment, it's supposed to show Walker's violence, and to balance culpability for Lark's actions between the IMF and the CIA, but in hindsight, Walker probably killed that informant in cold blood to protect his identity.
Another subtle foreshadowing: when Ethan questions Walker about his brutal methods, Walker rebuffed him by reminding him about the IMF's catchphase "Your mission, should you choose to accept it". Later in the movie, Solomon Lane gives Ethan an Armor-Piercing Question by asking whether there is a time that Ethan chooses not to accept a mission. Seeing that Walker shared a pretty similar idea with Lane, of all people, is a clever tip that the two are in leagues with each other.
Walker's evidence for why Ethan Hunt must be John Lark makes sense... until you realize that it requires information that Ethan didn't know, but Walker did. Most obviously, Walker claims that Ethan must have counted on "a neutral witness" (himself) being present to see "Lark" die, but Ethan had no idea he'd be there prior to the meeting.
This is also probably why Sloane is "in" with the IMF's ploy to expose Walker later in the film: she notices that something was off in Walker's report, and that's probably the reasoning that Walker is used as a "neutral witness" when Ethan couldn't have planned for because he didn't know Walker was going to go with him until Sloane intervenes. This, plus the overall the aggressive nature of Walker's accusations of Ethan being John Lark, lead to Sloane realizing that maybe her trusted right-hand man might not be as truthful as she thinks. Of course, this also leads to Sloane deciding to arrest everyone. She doesn't have faith in the IMF in the first place, and when her trusted agent might be plotting behind her back as well, she doesn't know who to trust, so it's safer to abort everything and find out who's The Mole first.
Julia's husband Patrick is a Nice Guy who is extremely friendly and accommodating to Ethan. When alone, when she's not looking at his face, he betrays his slight concern. He's not too stupid or oblivious to realize there might have been something between Ethan and Julia. He's just too mature to start acting jealous and overprotective. Julia obviously has good taste in men.
Part of The Reveal is that John Lark is fulfilling some demands and scheme of Lane, but it's hinted that Lane has control over the Apostles despite being in prison much earlier. Lark is set up simply as a sole extremist with one goal, having hired the "terrorist for hire" Apostles to deliver him the plutonium. The Apostles meanwhile? They set up a series of attacks, including the smallpox outbreak in Kashmir, which is revealed to be part of Lane's ultimate revenge on Hunt. Lane always plans a few steps ahead of everyone.
On a related note Walker being revealed as Lark can feel weird, since Walker screws up immensely in the first act of the film, from botching the HALO jump, wrecking the mask machine, panicking when Ilsa nearly shoots him. Not the kind of thing a mastermind would do. Except Lark isn't one. The exposition dump establishes that Lark is simply an extremist actor, having hired the capable and coordinated Apostles to help him. Walker's a highly trained special operator and assassin, not a spy or director. He is, after all, a "hammer." If not for the Apostles and Lane's skill, Walker / Lark wouldn't be able to do much of anything.
Speaking of him sabotaging the mask making machine, that's more than carelessness, considering his vocal skepticism about how well they work it can be seem him as openly disrespecting the technology. It's also something that ends up biting him in the long run: since Ethan never makes the mask, he never sees the technology in action and never realize how good the face and vocal technology are, and because of it he ends up being easily deceived by it.
Similarly, Lane is Ethan's Evil Counterpart. Like Ethan he's a "scalpel," a strategist who sees the bigger picture and details and how one action can spill over. Walker, or Lark, is Ethan's foil, a "hammer" who, ideology aside, works as a CIA Special Activities Division assassin. For him, anything is simply "hit the target." His big scheme? Set off nukes and hope the suffering causes people to overthrow the old order. Lane, even when planning to die in the explosion, realizes it takes a lot more than simply killing people. In Rogue Nation the Syndicate was carefully selecting targets and making sure actions caused reverberations and had the right fallout, not simply that they killed people. He's probably the one who chose the target for the nukes in Fallout, since Lark doesn't seem to think on the right scale.
Figuring out that Walker is deliberately framing Hunt and by extension, that Walker is Lark, becomes easy when one realizes that the phone given by Walker to Director Sloane is fake. The phone implicating Hunt was, while similar to the one taken from the fake John Lark, undamaged. The phone that Hunt and Walker find in the bathroom was heavily cracked due to the violent fight.
On second watching, a lot of things August Walker did in the first half of the film can be seen as him deliberately trying to sabotage Ethan's mission. He uses the mask-creating device to smash "John Lark"'s face and breaks it, forcing Ethan to go meet the White Widow without a mask and expose himself. He may have deliberately sabotage Ethan's getaway bike after they escape from the van so that Ethan would be chased by the police for a big while, giving Walker a lot of time to go to the safehouse and free to set up his own plan with the Apostles for further ambushes without anyone noticing. He deliberately chooses the crampy old BMW as their second getaway car so on the off chance they're still being chased by the police or someone else, they'll have a harder time to get away.
Lane's plan to escape IMF custody was a sublimeBatman Gambit; it's stated repeatedly that the CIA finds the IMF's Gambit Roulettes of masks, cons and acrobatics exasperating — they let the fox in the henhouse in the first place by assigning Walker to the mission, outright ordering him to screw up any plan Ethan comes up with if at any time he considers too complicated. All Lane needed to do was let the plot drag on long enough for the CIA to say "hell with it", whereupon they would attempt to drag them all back to Langley to sort the whole mess out at their leisure and his moles could strike with impunity.
No wonder Ethan didn't fight Lane hand-to-hand in the previous movie. He's about as good as Ethan.
Lane deliberately chooses the location of the first nuke to be Hidden in Plain Sighton purpose. He gambitted that the IMF would find it first and therefore they had to split up so that someone would remain to disarm it. The second bomb is deliberately well-hidden so the IMF would have to split up again to cover more grounds. After seeing Ilsa alone, he lets himself be seen by her to lure her into the house and get the drop on her easily while she's alone - then uses her to lure Benji there and do the same thing to him - systematically taking down Ethan's team one by one all by himself through divide and conquer.
The Reveal that the White Widow was actually working with the CIA makes sense when you remember that all the way back in the first film, her mother Max ends the film making some sort of deal with Eugene Kittridge to keep herself out of prison. Since Kittridge is a former director of IMF and the IMF is a branch of the CIA, it's reasonable to believe that whatever deal Max made with the IMF/CIA also passes on to Alanna as well.
Mixed with Fridge Horror: the Apostles' final plan could have done much more than poison the water. The Kashmir region is infamously in dispute between India and Pakistan, two nuclear powers who've been at odds with each other for decades. Nuclear detonation in that area would have probably ignited tensions and led to global thermonuclear war.
Fridge Logic
The movie ends up making Walker a subtle non-action guy. The first testament as his assassin skills are people who could have exposed Lark's (his) real identity, making it likely that at least a few let themselves be killed. Lark might be a hammer to Lane's scalpel due to the simplicity of his plans, but he is presented as a thinker first, a theorist through his manifesto, rather than someone who took direct action. All in all, Lark seems like a phisically fit planner whose assassin persona is forged rather than the foil for Ethan he is initially introduced as. This is true particularly coupled with the reveal of Lane's physical prowess, which makes him an actual foil for Ethan. The reason this is here and not under Fridge Brilliance is that of course this doesn't appear to be actually intentional from the movie, just a result of making Walker's skill an Informed Ability by giving two bad fights (first against possibly the best fighter in the franchise, and second requiring him to job quite a bit against the director because God forbid an authority figure could be anything less than a real badass in hand-to-hand fighting, it's just funny how it ended up still making a coherent narrative by accident.
