When Walker saves Ethan in the fight at White Widows palace, no mention is given as to how he managed to get past the security without causing a scene outside. On rewatch, knowing that hes 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.
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.
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.