Not everybody's daddy is the deadest shot in Maycomb.
The mad dog disease symbolizes racism in Maycomb county and most of the country during that time. Atticus 'battling' the mad dog and its diease, is similar to him battling the rest of the county when hedefends Tom Robinson. Everyone calls him a [Black Person]-lover, just for giving a black person a fair chance in court. The disease is spreading, and everyone has it because you get attacked when you do not agree with the most popular opnion. Atticus not feling proud of his talent for hunting, since he hates hurting things naturally more vunerable than him, is alike to his view on the degrading of black people. Black people are disadvantaged by society, and Atticus believes it would be wrong to not defend someone for their skin color and vulnerability.
Zeebo, Calpurnia's son, also does a cleaning job for a prominently white neighboorhood and their locals. African Americans were commonly cleaners for white people. They do the 'dirty' work.
Scout does not understand that Atticus did not tell them about his past for a reason. Her age makes her less aware than Jem, who already realizes Atticus does not want them talking about it in school.
I think the theme is to not follow people who hurt and degrade others who are vulnerable and defenseless, even if you would be the only one against it. You should always be a voice for the people made weaker by society, if that is what you stand for.