In general, Fitzgerld portrayed characters in his novels and stories who were very idealistic and passionate. They believed in romantic ideas like love at first sight and the integrity of an honest man. These two words hope and dream are key concepts in the novel, especially for Gatsby. He had to believe his dream of being with Daisy would come true, and so he worked tirelessly and key his hope alive. Daisy is a dream, a fantasy, ad Gatsby's desperate, fervent hope is what brings him to her again, via his hard work to become a self-made man of wealth and status and allows him to win her love. Daisy also always hoped she could be with Gatsby again, but the dreamlike quality of their love is revealed when it becomes clear she is not willing to change her life to be with Gatsby alone.
Fitzgerald also contrasted his hopeful characters with the idea that fate and destiny intervene and that bad luck can trump the best intentions. The death of hope and dreams is also a common theme; the loss of youth, the disintegration of love and ambition, these were ways that the hopes and dreams of young characters could be shattered.
Fitzgerald uses the words "hope" and "dream" so often because these are the concepts that define the titular character, Jay Gatsby. It is his dream of being reunited with Daisy and his hope that she will still love him that motivate him to earn and spend and live the way he does. Nick describes Gatsby as having an "extraordinary gift for hope" -- he believes in one's ability to relive the past, something most of us think is impossible. And it wasn't Gatsby's dreams that ruined him in the end, it was the "foul dust" that floated in the wake of those dreams; it is, in part, the other people who are less idealistic than he is, the people who cannot bring themselves to dream or hope as much as he does, that ruin him. They make the world an impossible place for those who dream and hope as Gatsby does, and it is this impossibility that Fitzgerald seems to mourn the most.
As noted in the question, the words "hope" and "dream" appear numerous times throughout F. Scott Fitzgerald's The Great Gatsby. One reason these words appear numerous times is the theme of the pursuit of the American Dream. Both "hope" and "dream" exist as necessities within this ideal. One in search of this would hope and dream about a better life. The novel speaks to a time of dramatic change for women and a time of exuberant celebrations (the Roaring Twenties). One's hopes would be that his or her dreams would come true.
For many of the characters, each is in search of their own American Dream. Gatsby is the iconic "self-made man" who fails to see his real success based upon the fact that he cannot have the one thing he desires: Daisy. Materialism is central to the "dream" in the novel. Wolfsheim's dream forces him to gain money criminally; Baker cheats to obtain her dream; others victimize, betray, and murder to obtain and keep their dreams alive.
Without the repetition of the words "hope" and "dream," a reader may not understand just how important the concept of the American Dream really was to the characters. It is through the repetition of these words that drills into the reader that this is what the test is really all about. Fitzgerald is only illustrating, through this repetition, how important success is in obtaining one's dream really is.