I bet what happened in the omake was in fact an act of senpai to protect kanako from any dangerous relationship with the police. And he explained that to her later.
But it's just a guess.
I feel like I missed a chapter. The cat's out of the bag, and we can guess with the flashback what was Rune's reaction. But about the Lady, it's not that clear.
What's all that deal with the title 'mister'? It's not like he was called grandpa or uncle or similar. And is that guy obsessed with staying physically young?