In "The Black Cat," why does the narrator mention the wall to the police?
The narrator in "The Black Cat," by Poe, mentions the wall to the police because he can no longer hide his black deeds, and he must be punished for those deeds. The foreshadowing for this punishment begins at the start of the story when he hangs Pluto, and the cat's image with a rope around his neck is emblazoned on the wall after the fire. This imagery continues with the white spot on the second black cat's breast that resembles the gallows. From the beginning of the story, the narrator experiences a decline and ascribes his terrible deeds to the use of alcohol though the reader knows that he has truly become an evil character.
He begins by loving Pluto, the first black cat, but, as his temperament alters, he finds himself in states of rage. First, the narrator takes Pluto's eye out, and later he hangs him. This hanging is the first foreshadowing of the narrator's own fate. Upon getting the second black cat which, ironically, is also missing an eye, his dread and rage at the new cat continue. Ultimately, he kills his wife and unknowingly walls up the live cat with her in the cellar.
The narrator's bravado (ego) leads to his rapping on the wall. However, more importantly, he must pay for his crimes against nature. Thus, the explanation of the imagery of the gallows at the very onset of the story predicts the narrator's own fate and the sin for which he must pay.
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.