Does understanding a culture help one to sympathize with the characters and why?
This is actually a complex question. One purpose of English literature through the ages has been to reveal the culture of the author's era to readers. Think of Charles Dickens and any of his stories, for instance "A Christmas Carol" or David Copperfield. These are written to show his contemporaneous readers the weak spots in their culture, as they lived it day by day. One of these weaknesses was that of valuing money and monetary gain over and above human relationship and mercy ("A Christmas Carol"). Another was the immoral consequences of making children, women and men, indeed whole families, suffer because of "pecuniary difficulties" (money trouble). Thus, historically, literature has been written to open the eyes of readers to cultures imperfectly understood.
On the other hand, while literature is meant to deliver understanding of the culture being written about (as in Farming of Bones, Edwidge Danticat), pre-existing knowledge of a culture can help the reader find a deeper comprehension of the authorial message. Thus, if reading Shogun by James Clavell, reading about Japanese culture will help deepen understanding of the fine points of the culture. An example is knowing the background, history and meaning of the Tea Ceremony will add dimensions of understanding to the characters psychology (cognitive and emotional states) and motives (reasons for saying or doing what they do).
So, yes, even though part of the task of literature is to reveal an imperfectly understood culture to a reader's comprehension, knowing about the culture being written about can help the reader sympathize with the characters because fine points of meaning will be more readily understood.