In regards to the nearly empty vial left in Bellatrix’s cell:
Back when we first saw the flask, I remember there being confusion over the point of leaving something foreign behind what was supposed to be “the perfect crime.” It definitely came in handy once they were found out, but it didn’t make sense to leave it behind when there was a serious possibility that they could have gotten away with it. I was thinking about the arc the other day when it (belatedly) occurred to me just what Quirrell might have been thinking.
Just before this point, we are told that in the MOR-verse, a prisoner stays in Azkaban until their sentence is up, even if they die. For those with life sentences, their corpse stays there until they need the cell. Furthermore, snake!Quirrell confirms that she’s alone, and thus is the only live prisoner in her particular area. Therefore, it would be reasonable to say that after Bellatrix died, the Aurors would stop frequenting that particular area until the next time someone who does “worse than murder” needs to be locked up. Even then, it’s possible that there are other empty cells, or (more likely) that the guards dispose of the older corpses first, for the symbolism if nothing else. So it seems probable that even after the guards return to frequenting that corridor, Bellatrix’s cell would probably be left alone the longest.
Additionally, much ado has been made over the destructive powers that Dementors have on their surroundings. And Bellatrix, being considered the most evil person in Magical Britain after Voldemort “died,” is in the very worst location, in close proximity to the nest of over a hundred Dementors.
Moreover, Dumbledore makes a point of stating that the titanium bars of the cage of the Dementor that is brought to Hogwarts were trueforged, since they would last longer than if they were created magically. He also notes that while the Dementor would start to degrade the cage, it would take longer than a day, possibly implying that the destructive field is strong enough to make the destruction of titanium within the course of a few hours a serious concern.
All of this leads me to believe that Quirrell’s reasoning was that if they had managed to pull off the perfect crime, the guards would have discovered “Bellatrix” dead from Dementor exposure in the morning. I don’t see the guards investigating a cell very carefully—or at all, really—when a prisoner dies in such a natural way, so the vial being put in the corner and covered in the strip of cloth that “nearly blend[ed]” in with the wall would keep them from noticing that anything was amiss in a cursory glance. After that, the cell would have been left alone for a long period of time, very near to the majority of Dementors, with the flask possibly being magically created as well. In all likelihood, this would have resulted in there being no traces whatsoever by the time anyone else would have entered the cell, preserving the perfection of the crime.
And if the crime wasn’t as perfect as Quirrell thought? Then they have a clue left behind, one with all the earmarks of Voldemort, neatly diverting suspicion away from Harry.
Well, that’s certainly one way to explain away all of the strange aspects. Establish them as fact, through the mysterious bond between LV and HP, and do so in front of a huge crowd so that the word can spread and mutate on its own. By the time anyone comes to investigate or question, they will already be influenced by the show or rumors they’ve heard, promoting that hypothesis to their attention rather than coming to it naturally.
It’s pointing the police at Mortimer Snodgrass, from chapter 17, as it were.