You have a lot of questions about the story. Usually it's best to ask one question at a time, but I shall try and get you started here with a few ideas about some of your questions.
The setting of the story is Paris, "in the summer of 1940," and, more specifically, much of the story centres around one room in a house "in the street called the Rue Morgue." In this room, an old lady is brutally murdered in suspicious and perplexing circumstances. The door to the room is locked from the inside, and all of the windows also appear to be firmly shut and locked from the inside.
The main character in the story is a man called August Dupin, who acts as the detective in the story. Part of his characterization is that he is depicted as possessing a "busy mind" and "an unusual reasoning power." He is also described as having a manner which could be "cold and distant." August Dupin is essentially the precursor to Arthur Conan Doyle's character of Sherlock Holmes. Although Dupin is the main character in the story, the story is narrated from the point of view of Dupin's anonymous friend and assistant, who is a precursor to Conan Doyle's Dr. Watson.
The plot of the story follows a now familiar detective genre plot. We begin with a mystery, namely how the woman in the aforementioned room is murdered. We then receive various clues from eye-witnesses and from Dupin's skills of deduction. Some of these clues are red herrings, or false clues to lead us away from the real answer. At the end of the story we then discover this real answer, and all of the questions we had previously are answered or resolved.