Let's play.
It is indeed a staple of classic narration to have the guy who is trying his best to be straight (in more ways than one!) shoot unarmed people in the face...
Maybe you do that after a thinly veiled death threat?
Maybe because Osep was hoping he would stay to run the clubs and maybe because the suspicion materialized in the form of a goon that took a bullet in the head?
Yeah when they dropped 10 names of characters that had 2 minutes of screen time each if at all I really felt like they went too far with the hand holding...
Maybe because he got some radio info from the guys remaining in the underground tunnels? Maybe he himself tailed Paul in the tunnels and took another exit when it was clear where Paul might exit? We don't know and we will never know because, once again, the type of narrative focalization used will result in huge chunks of information the viewer will never get access to (which also explains while several of these chunks are made clearer through dialogue, because, by choice, they cannot be shown).
As far as the document being inadmissible in court... who cares? That really would be a problem if they were in a situation where they knew everything but needed evidence to incriminate people. They were in a situation where they had very little idea of what was going on and how people where linked with each other and these documents proved extremely precious to establish that. They will not be admissible in court, but now, at least, they know what is going on and are in a much better position to work the case and find witnesses and evidences. Well... they were before people started dropping left right and center that is!
The double list is probably just a prop blunder by a director that does not realize HD is HD and forgot people can and will press pause whenever there is a note of some sort on screen. I can hear it from here: "That list looks too damn short! Fill this page with lines so we can see something big is cooking! 5 minutes break until we get the new prop!" On a side, the shopping list make it look like a 2 men job... Seymon and Nails? Or some Bonnie and Clyde action with him and his wife?
Lastly, Vaughn was indeed better, but mostly because his lines were better and the character is no longer holding his horses so tightly. Even when, he got verbose with the diamond seller it worked well in a super passive-aggressive 'I am taking none of your hypocritical shit motherfucker' kind of way.
I think my point was, when writing a show, you simply demonstrate your meaning in the things you're writing and showing on-screen instead of just randomly having shit happen for the fuck of it or because it's cool and hoping people with throw out a ton of 'maybe' defenses to explain things. It really wouldn't be that hard, and in plenty of instances would have heightened the suspense, tension and drama.
For instance, instead of just having Lt. Burris inexplicably standing by an exit and leaving us to guess if the Captain was just knocked out and phoned him for help, you show, get this, the Captain waking up and phoning for help. Then, instead of a random threat we don't know about suddenly gunning down a main character for cheap shock value, you actually have a main character potentially walking into a known threat so you can worry about him. Too often the show settles for cheap thrills via inexplicable unexplained action instead of actually building suspense and tension by letting the threat be known.
Yeah, exactly, another prop blunder. In a show that has felt rushed all around. Like they haven't had anyone editing or even beta reading the script. Like the editing sometimes makes no sense. Incongruities like the big fight scene where the final raid boss gets show and falls on his back, and then the next cut he's laying face down, then he's suddenly on his back again. On an objective, nuts and bolts level this show has been a mess and is clearly suffering in quality on just about every level, and instead of just recognizing that as fact, people are vehemently defending the show and shouting down any detractors as if there haven't been a ton of quality issues on all levels.
And the editing really is shitty. Fucking distracting shitty for those cursed with noticing that kind of thing. Breaking scenes right in the middle of a conversation to cut away to other action, only to return right back. The opening scene where we got the relatively boring, calm scene of gaycop looking over contracts, implying they've settled into a motel after some crazy shit, and then we get scene where Ani is flipping out like they just got into the hotel and she's having a hard time decompressing. Very simple fix to just switch the sequence of those scenes which not only demonstrates a more plausible passage of time and events, but starts with the more interesting scene more immediately (sorry gaycop but contracts are boresville). And if you're showing a series of events like Frank setting shit on fire, you intersperse other scenes to give a sense of time passing, which they started doing, so you understood he was driving around setting the town on fire, you don't have him lighting shit on fire in one scene and then immediately cut to him on top of makeout point overlooking the city (unless he can teleport). Just shitty, shitty overall editing and pacing and scene choice and logical flow; again because it's evident this season is very rushed and nobody seemed like they were prepared.
Admittedly, most people don't care about this craft-level shit because it doesn't change the fact they're excited to see an explosion or sex or a shootout, and whether it made complete narrative sense or not doesn't change their 'that was cool' titillation. So, no, I don't give a shit if 'maybe' somebody in the tunnels called someone else or any of the other 'maybe' explanations because it would be pretty easy to just write something that makes sense so we didn't have to discuss dumb shit like that and could instead discuss why the only people sad that gaycop died are the ones that like inexplicable shootouts.