Are Coppola and Coppelius really the same person? Is either one of them really "the Sandman"?
It seems by the end of the story that Coppola and Coppelius are, in fact, the same person. This can be gathered from the scene in which Coppola runs off with Olimpia's body, after which Spalanzani calls Coppola "Coppelius" while mourning the loss of his automaton/daughter. Whether they collectively are the Sandman is a bit more complex as Nathanael's idea of the Sandman evolves multiple times through what people tell him they believe the Sandman to be and what he learns from personal experience. If the Sandman can be taken to mean an evil person seeking out eyes, Coppelius/Coppola do seem to fit that description. However, they/he seems to be purely human, after eyes so that he may study them for use in their automatons.
How does Nathanael's love for Olimpia influence the trajectory of the story?
Nathanael's love for Olimpia shows two things - that he is able to control and warp his own perception of people and events and that he has problematic views of the ideal woman. That he is able to convince himself that Olimpia is alive, even after seeing signs that Olimpia is not alive like her lifeless eyes and cold hands and lips, makes the reader question whether they should trust his perception of Coppelius and other events. His view of the ideal woman as one that can only move gracefully, listen politely, and say a single word of response is parodied by Hoffmann when Olimpia is discovered to be an automaton and other married couples are forced to allow women to speak and yawn as well.
Why does Hoffmann use a narrator who is not directly involved in the story? What effect does this have on the reader's experience of the story?
That the narrator is not involved in the story means that the reader is unsure of what facts and descriptions to believe. The narrator clearly loves storytelling, which already sets up a potentially unstable situation. Furthermore, he seems to be getting his information from Lothar, who has a biased view of the situation. Finally, the narrator seems very affected and knowledgeable of Nathanael's perceptions of events, sometimes seeming to blend them into the narration itself and causing question of who exactly the narrator is. The effect in all cases is the sense of the fantastic, in which the reader is unsure of what elements of the story are reality, fantasy, and/or insanity.
Why does Hoffmann include the scene of Clara's domestic life at the end of the story?
It seems that Hoffmann's ending of the story with a scene of Clara's later domestic life gives the story a moralistic edge. Nathanael is seemingly punished for a number of things - his insane obsession with the Sandman and his switching of love from pleasant Clara to falsely perfect Olimpia being the main two. Showing that Clara has been able to move on with her life punishes Nathanael for these acts, sets up Clara as the main secondary so that one may look back on her plot line more carefully perhaps with regard to Hoffmann's parody on the position of women, and demonstrates the ability of life to move on sensibly after madness and tragedy.
What do eyes symbolize add within the context of the story?
Eyes are used in the story to a number of ends. First, eyes are a major way in which people are supposed to be able to determine whether someone is living and even see into their soul. This seems clear to all characters in the story, causing Coppelius to focus so much on eyes so that he can perfect them in his automaton and causing others to question Olimpia (though Nathanael convinces himself that he does see life in her eyes). Eyes are also an important sensory organ for perceiving the world around one, and this underlines Nathanael's and the reader's inability (and the fear that comes with this inability) to trust what is shown. Finally, eyes as a historically religious symbol adds to the tone of religion and dark magic woven throughout the story.