Let's play.
The same way you have a gun on 4 guys about to kill your ass who all willingly just dropped their guns and instead of killing them you run away and kill them in far more contrived ways?
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...
The same way the female runaway who is actually not at all sympathetic to you 'saving' her is sent away with her sister despite being your best witness to back up your case?
Maybe you do that after a thinly veiled death threat?
The same way you let the guy you just fucked over by ruining his entire empire continue to run your clubs and then aren't at all suspicious when an hour later he has the casino evacuated?
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?
The same way everyone constantly tells their entire motive, history and plans like they're in a Bond movie or talk about the things that everyone already knows as a means of reminding the viewer of things that happened to ensure you understand?
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...
How did he know gay cop was coming out that exact spot?
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.