Carol and Peter had one date. The reason why it was awkward was because they were not being themselves as they were trying too hard to impress the other. They bonded over junk food, which is Carol's personal favorite. While I do not expect Carol and Peter to be a romantic couple, I could see them as close friends, along with Jessica Drew. What would be cool to see is Captain Marvel, Jessica Drew, and Peter Parker working together in a story arc. Let see their combined talents to the test.
As for Carol and Rhodey, if they have so much in common, why did they breakup instead of working things out like Peter and MJ? Rhodey and Carol should work out their problems instead of running away from them.