Where do we see Amir struggle with the conflicting Afghan and American cultures? Why does Amir seem to be reluctant to completely move on from his Afghan identity?

To answer the second question first, Amir is reluctant to move on from his Afgani identity because he finds he cannot escape his past in Afghanistan, even though he wishes to do so. As a child, Amir betrayed his friend and family servant Hassan. Later, when he and Baba move to America, Amir hopes it will be a fresh start, but he finds that he can never forget what he did to Hassan. It's also true that Amir grew up in Afghanistan and that this country will always in some way be his home. He does also have positive memories of his childhood there that he eventually wants to recall. When he returns to Afghanistan as an adult, he revisits the house where he and Baba once lived, and when Farid advises him to forget and move on, Amir tells him he doesn't "want to forget anymore."
Amir's struggles between America and Afghanistan have mostly to do with the memories of Hassan and Ali, whom he left behind in Afghanistan, and both are dead by the time he returns to his home country. Amir does not seem to have much trouble adjusting to life in America, unlike his father, who must take a low-paying job after having been a respected business man in Kabul. Amir graduates high school and later college and becomes a published author, as he always dreamed. He has a happy marriage with Soraya, though that is tempered some by their inability to have children. It is more difficult for Amir to go back to Afghanistan when Rahim Khan calls him to do so than it ever was for him to go to or live in America. When he returns to Afghanistan, it is a much different place than the land of his childhood, though the violence and turmoil began when he was young and he and Baba fled to Pakistan. He must face the criticism of Farid because he (Amir) had the means to leave and hasn't suffered through the worst the country has seen and felt. He must face the brutal reality of the Taliban's control and even has to fight Assef, now a Taliban leader, and deal with many obstacles on his way to adopt Hassan's son Sohrab and bring him back to the U.S. Then we see Sohrab's trouble adjusting to life in a new country after all the trauma he has experienced in his young life.