1 have you been
B yes I have already seen
2 has he ever ate
No he has never had it
The given statement is an example of a phrase, specifically, a participial phrase. The element that is necessary to complete the sentence is an independent clause. A participial phrase consists of words with the -ing or -en form plus its modifiers. This kind of phrase always functions as an adjective. The phrase "Beginning in the fall of 1957" is describing a certain noun, which is found in the independent clause.
Edgar Allan Poe uses chronological events in "The Black Cat" in order to show the curve of transformation the narrator suffers through the story. At the beginning he used to like the cat very much, but after certain events that happen to him and that lead him to commit certain actions, he starts hating the cat.
Without the chronological events, the story would be very difficult to understand and we wouldn't be able to understand his internal motives to do something like that to the animal he used to love.