Gatsby's dream is basically the American Dream. That is, Gatsby wanted to make himself into a rich man who could have not only wealth but also love (Daisy).
The second half of Chapter 8 shows the results of Gatsby's dream. Gatsby, of course, is murdered by George Wilson. Wilson mistakenly believed Gatsby was responsible for the death of his wife. What Wilson didn't know, however, is that Daisy was driving Gatsby's car; therefore, Daisy was responsible for her death.
Therefore it is ironic that Gatsby's love for Daisy led to his death. His dream, then, is ultimately unattainable. He will never achieve the happiness he hoped for, and the person he loved most brought about his demise, albeit accidentally.
The general insinuation is that Scout is too boyish for the times they live in, so Aunt Alexandra decided it would be good idea to teach Scout how to be a lady while she (Alexandra) was staying in town.
Belonged is your possessive noun in that sentence.