What is the purpose and subject of Paper Towns?

John Greene’s novel combines elements of a mystery, love story, and hero’s quest. The protagonist, Quentin Jacobsen (known as Q), sets out on a quest to find his beloved old friend, Margo Spiegelman. Nourishing an unrequited love for her since childhood, initially the teenage boy is concerned about Margo’s increasingly volatile behavior, including pranks that have become criminal acts. When she disappears, however, he and some other friends go looking for her.

Paper Towns is also a coming-of-age story, in that Quentin must come to terms with his idealistic, romantic, and gender-biased worldview. When located, far from being relieved at being found, Margo lambasts Quentin for his antiquated ideas about male-female relations: he has been living a fantasy in which men are knights who rescue damsels. As it seems he will adjust positively to this change, the reader can see that Quentin has grown through the experience despite its unanticipated outcome.

Paper Towns is a deconstruction of the "manic pixie dream girl" trope. Margo Roth Spiegelman, the leading lady of the novel, is an adventurer who seems perfect and wild in every way. Very few people know anything about her. The narrator of the novel, Quentin, believes that Margo is a "miracle" and spends most of the book viewing her as an answer to his problems rather than as a person. The book is about Margo's disappearance and Quentin's determination to find her, but more than that, it's about Quentin and Margo both finding themselves. The original setup looks like Quentin is going to find Margo and "get the girl," but at the end it turns out he knows just as little about Margo as Margo knows about herself. He realizes that he's been disingenuous about Margo's personality by viewing her as more of an idea than a person, and also realizes that he has to let Margo live her own life and be a real person.

John Green has stated that his purpose in writing Paper Towns was to rectify something he had not accomplished in Looking For Alaska—creating a troubled female character who stood on her own rather than being a plot device for his narrator's development.

What is the theme and topic of Paper Towns?

Your question is interesting because, in fact, both "theme" and "topic" can often be used as synonyms of each other.  In regards to "Paper Towns," the main theme is one of superficiality (both in love and in hometowns).

Why is the theme one of superficiality in love?  Because the main character, Quentin, spends his entire young life being in love with his first sweetheart, Margo, but finds out during the course of the story that she is not the person he thought she was.  There are clues throughout that this is the case.  Quentin is intelligent and less likely to take risks while Margo is "cool" and more likely to take risks.  Take, for example, the night when Margo shows up at Quentin's window before high school graduation.  Margo has a night of revelry planned.  Quentin hopes it will cement their relationship, but Margo leaves, requests to be found, and Quentin finds himself while pursuing her.  Ultimately, he finds out that Margo is not for him.

Why is the theme one of superficiality in hometowns?  Well, this story is about the challenges in the culture of central and south Florida.  There is too much development, too much heat, too many temptations, too many cookie-cutter neighborhoods designed to eventually fail. These are the "paper towns" that are meant to fall apart.  Quite true in my own experience in growing up in South Florida.

Thus, you can see the main theme or main topic of the story is certainly superficiality in relationships and towns.  Quentin finds out the truth of this main theme during the course of the book.