In chapter 4, why does Gatsby have Jordan ask Nick if he'll invite Daisy to his house, instead of asking Nick himself?
Gatsby, for all of his money and seeming popularity, is also very insecure. He hides in the shadows, lurking and watching from afar. Think to the scene when Daisy does come to Tom's. He is ready to give up before she is even late arriving. He has put so much effort into being the man he thinks Daisy will be able to love, yet he does not love himself. He expects to be shot down before he even tries. He sees Jordan as a safe intermediary. It is, to put it in modern terms very much "high school" behavior having someone else arrange a chance encounter for you. It is a sign that Gatsby lacks a certain level of maturity and confidence. He may fake it quite well, but he is not as sure of himself as he would lead Nick to believe.
I think this continues an unfolding revelation about Gatsby's overly confident inability to have confidence. Gatsby is somewhat of a paradoxical character. He carries himself with such great prestige and that "old sport" business makes him everyone's best friend. But, he really struggles with personal relationships.
Another reason he might have asked Jordan is that she appears to have something of a relationship with Nick. Maybe that made Gatsby feel like he had an in if he went through her.
I think that he does it for a couple of reasons.
First, I think he might be a bit embarrassed to do it himself. He might think it would be easier if Jordan did it so that he didn't have to ask such a personal favor himself. She says Gatsby was afraid Nick would be offended.
Second, I think he wanted Jordan to tell Nick about Gatsby's past with Daisy. I think he figured Jordan could tell Nick about it and make Nick a lot more likely to be willing to help him out.