Things that don't make sense (v2)

Well, at least in ME1, every character HAS every weapon on them. In the later games, Shepard routinely conjures up assault rifles in cutscenes even though s/he doesn't even carry one.

There's no point, because they maintained that Sovereign was a Geth ship and not a Reaper. I mean, they literally had it's corpse and still denied that it was a Reaper, so a recording wasn't going to change their mind. Besides, during the Citadel DLC we find out that the council did acknowledge the Reaper threat a long time ago but for some reason ignored Shepard's warnings - just one more reason to leave them to die.

Here's another one: Garrus makes a perfect shot on the men threatening Dr. Michel... and then never fires his gun again while Shepard and his team take out the remaining 3... even if the player delays and delays doing it.

The wrong gun appearing in cutscenes is one of my biggest pet peeves in all of gaming I think.......

I would guess it was found on Freedom's Progress. 

Maybe it was collected by another Cerberus team? We know the illusive Man had spies placed in several of the colonies, so maybe they were able to secure a drone?

I usually head canon that he's looking at a holographic image of one he's compiled from Veetor's omni-tool data; and the illustrator of the scene just messed up and drew it in like an actual living specimen.  Some of the dialogue doesn't quite fit with my head canon either, of course.

I can actually defend this point of view. But before I will preface it by saying that to an extent I agree more with you. That the council's denial of Shepard's testimonies, not so much from before the Sovereign Attack on the Citadel but after, was ludicrous.

However, the council's point of view on Shepard's testimonies, especially with Shepard's initial accusations against Saren, could have been seen as the council believing Shepard was making up stories to try and make humanity sound better, a political move I guess you could say. Not to mention the council member that really doesn't believe you is Sparatus, the Turian councillor. Whose denial of your claims also supports his own race from doing any wrong in Saren.

The council always just saw Sovereign as an advanced Geth ship. And when they're being attacked by no other foot soldiers than the Geth themselves. And to a lesser extent husks, which could be rationalized as Geth experiments. Why would they believe Shepard?

Also remember that the Citadel Council had been around for almost 3000 years before Shepard's accusation. So I'm sure there would be at least a touch of ego there, thinking they know all there is to know about the galaxy.