There is quite a bit to ask Nick Carraway, as he leaves a lot of information out of his narrative.
A question that pops into my mind is one that critics have posed: Nick talks near the end of the novel about an obscenity written on the steps of Gatsby's house at the time of Gatsby's death. Nick rubs it out with his foot, but we never learn what it says. In a letter, I would ask what the obscenity was, why Nick thought it was written, and if it seemed to speak to any truth about Gatsby.
Ellipses (". . .") are a punctuation mark that indicate that something has been left out of the narration. These also raise curiosity, as they occur frequently in the narrative. Much has been made of Nick riding the elevator with Mr. McKee and ending up in his hotel room. I would ask Nick what went between the elevator ride and the following ellipses:
. . . I was standing beside his bed and he was sitting up between the sheets, clad in his underwear, with a great portfolio in his hands.
What exactly was going on with Mr. McKee? Why does such a small character enter into the narration at all? More to the point: is Nick communicating to us that he is gay? He couldn't say this outright in the 1920s, but he could today. I would also ask if Nick believes or knows Jordan was gay and was simply using him as a cover story.
I am also curious about what happened with the girl back home in Chicago: we never do get any definitive closure on that episode. Nick says he needs to break up with her: does this happen? What happens?