The fundamental difference lies in how each views the residents of Maycomb. Atticus views each family, like the Cunninghams, as an essential part of Maycomb. Remember, early in the text he helps Mr. Cunningham with his "entailment." And Mr. Cunningham pays him as best he can - not with money but with fire wood and nuts. Central to Atticus's beliefs is the idea of crawling inside someone's skin and seeing the world from their point of view (he often tells Scout to do this). Because he has the ability to see the world through the eyes of Mr. Cunningham, Atticus can find something in him to respect (for example, Mr. Cunningham will not accept charity - nor will any of his children). Atticus wouldn't classify the Cunninghams as trash. They make the best of what they have. Notice how he makes Walter feel at home early on in the book when Scout has him over for lunch. Walter and Atticus talk about farming and life. Atticus doesn't judge him nor does he look down at Walter.
Alexandra, on the other hand, lacks the ability to view people from any perspective other than that of being a Finch (which means she is of better stock than most residents of Maycomb). This is why she tells Scout that she can like the Cunninghams all she wants, but they are trash and they not to be treated as friends or invited over. She is much more interested in the social perception of her family than Atticus is. But also remember that Alexandra would never have taken the Tom Robinson either.
In a nutshell, Atticus believes in truth and honesty regardless of the social cost. This is what leads him to take the Robinson case in the first place. Alexandra, although she is humane and decent person, is more concerned about how their family is viewed in relation to other wealthy, Southern families. She thinks Atticus should not be defending a black man simply because that's not how good, white lawyers in the South should act.
To your second question, absolutely not. The Cunninghams are certainly poor and viewed by the community as trashy, but they are definitely not trash. Walter Cunningham is a proud man who doesn't accept charity and is strong enough in will to do what is just and right. Remember that he is the one who leads the lynch mob away after Scout spoke directly to him. It would have taken a ton of courage and pride for a man of lesser social status to stand up against the rest of a foaming mob that way.