In addition to his tales of rural life along the Mississippi River and the slavery issue that exists there, Twain also delves into the issues of superstitions, children's rights and parenting skills, among others. The issues of children's and parental rights are magnified when Huck's ne'er-do-well father is awarded custody of Huck, an act that removes the boy from the more proper family life that he shares with his aunt. Huck's father obviously has no interest in bringing up his son in any kind of normal household; he is only after Huck's money. Twain shows how average 19th century people still maintain their beliefs in various superstitions, and Huck exhibits his knowledge of several different forms during the course of the novel.
Along with the underlying issue of slavery, Twain spends a great deal of time describing and portraying the attitudes that people had towards not only slavery, but the very idea that certain groups of human beings required different laws, different attitudes, and were not necessarily compatible with everyone else. If you look at the chafing between Huck and his father when Huck learns to read and to educate himself, or the conflict between the widow and Huck as she tries to understand this obviously bright boy that doesn't always want to follow the rules, you see that a difficulty in understanding each other is a common theme, pre-civil war and now.
Another aspect that is portrayed that isn't obvious is the lack of government authority that existed in much of the South and much of the US at that point. The fact that so many conflicts and problems had to be resolved locally, depending on local resources and local intuition is a part of the country's history that is gone now with our centralized schooling, centralized law-enforcement, etc., etc. Twain brings out the way that people on the frontier had to work out their own problems both with their land, their making a living, as well as with other families or groups of people around them. The constant change led to all kinds of interesting problems but also a constant state of innovation and compromise evident in the story of Huck and Jim and their adventures.
I would say that the most obvious characteristic of America in the time before the Civil War that Twain describes in this book is slavery.
Much of the book centers around Jim's attempt to escape from slavery and around Huck's dilemma about what to do with Jim. This part of the plot describes how things were in the South before the Civil War put an end to slavery.
Along those lines, I would say that the description of the feud helps to show how Southern society was in those days with its emphasis on honor and such.