In "The Black Cat," why does the narrator mention the wall to the police?
You asked three questions - as an enotes editor I am compelled to reduce this to one question according to their regulations. This excellent short story by Edgar Allen Poe in many ways is similar to "The Tell-Tale Heart" - both feature an unreliable first-person narrator who commit terrible crimes, try to hide the body and then take the police to the very place where the body is stowed, and then have their crime revealed at the point when they are most certain of getting away with it.
In "The Black Cat," it is clear that the narrator feels he has found the perfect place to conceal the body to ensure that he will never be found out. Note what the narrator tells us about the search the police make:
Secure, however, in the inscrutability of my place of concealment, I felt no embarrassment whatever... The police were thoroughly satisfied and prepared to depart. The glee at my heart was too strong to be restrained.
It is this staggering sense of over-confidence and his delight of having committed the "perfect crime" that leads him to rap at the precise wall which concealed his dead wife's corpse, leading to his downfall.