"What happened down in the dungeons between you and Professor Quirrell is a complete secret, so, naturally, the whole school knows."Was Q sincere in 'Q Who' when he offered to renounce his powers and join the crew? Was he banking on Picard saying no, and what the hell would he have done if Picard had answered 'yes'?
Q is rarely sincere about anything.
I think it will never be explained why Kruge killed Valkris when the tape had become widespread knowledge.
The tape was not widespread at that point. As far as we know, Valkris was the only Klingon who had seen it. Kruge killed her to keep his mission a secret, and so he could take credit for bringing Genesis to the Empire.
Indeed. Starfleet makes a point of informing all the Enterprise crew that the whole business with Genesis is classified and not be discussed--and they even assign an undercover agent to McCoy to make sure he doesn't blab about it in public. (Remember the scene in the bar?)
We're told there's a controversy, but not how much the general public knows about Genesis, which Starfleet is trying (and failing) to keep hush-hush . ....