One particular incident in Chapter 19 shows something very important that Dill hears and understands, and it reduces him to sobs. After listening to Mr. Gilmer question Tom Robinson at the trial, Dill is sickened by the cruelty of Gilmer's treatment of Tom. The prosecutor bullies Tom on the stand, and Tom is powerless to defend himself by standing up to Gilmer. When Tom answers one specific question honestly, daring to disagree with Gilmer, the prosecutor slaps him down, demeaning him: "Are you being impudent to me, boy?" Tom immediately replies humbly, "No suh, I didn't go to be."
It is at this point that Jem makes Scout take Dill outside:
For some reason Dill had started crying and couldn't stop; quietly at first, then his sobs were heard by several people in the balcony.
Once outside, Dill tries to pull himself together, then explains to Scout and Dolphus Raymond nearby why he had become so upset: "That old Mr. Gilmer doin' him thataway, talking so hateful to him--" When Scout tries to explain that Gilmer was doing his job, like Atticus, Dill explains further:
Well, Mr. Finch didn't act that way to Mayella and old man Ewell when he cross-examined them. The way that man called him "boy" all the time an' sneered at him, an' looked around at the jury every time he answered-- . . . . "I don't care one speck [that Tom is a Negro]. It ain't right, somehow it ain't right to do 'em that way. Hasn't anybody got any business talkin' like that--it just makes me sick.
In the courtroom, Dill had heard the ugly voice of racism and understood it for the evil it is. The injustice and cruelty of it made him cry and made him sick. He understood Tom's fear and powerlessness, and he understood Gilmer's arrogance and persecution.
One incident that shows Dill's hopes occurs when he runs away from his parents' house and travels many miles back to Maycomb, showing up late at night, tired, dirty, and hungry. Dill comes back to Atticus's house because it is the only place he thinks of as home, the place where he as found acceptance, affection, and respect. He hopes desperately that he will be allowed to stay.
If you're looking for one specific important incident which covers all of the criteria, it would have to be during the trial.
Throughout the trial, Dill hopes Tom Robinson will be cleared especially after Atticus gives the jury several reasonable doubts. However, when the prosecutor gets up and begins to hammer questions at Tom until he admits feeling sorry for her, Dill cries because of how unfairly Tom is being treated. When he hears the guilty verdict, he understandsit means Tom Robinson, because he is a black man, is going to jail for the rape of Mayella Ewell.
If you're looking for separate incidents then let's begin with cries. Again citing the trial, Dill cries when he witnesses how unfairly Tom is being treated because he is a black man.
Dill understands more than people give him credit for. He gets that Scout sees herself as an equal to Jem and he also gets that she doesn't understand the confusion she feels toward Jem and his adult behaviour. Their conversation in bed after he's run away shows that he has a greater understanding of other things around him especially when he makes the comment about Boo Radley having no place to go.
In the first chapter, he hears the stories of Boo Radley and decides to make Boo their summer project by trying to make him emerge from the home he has been imprisoned in these many years.
He has a variety of hopes. One of his greatest revolves around the fact that he wants his mother to notice that he is still around. This lack of attention is the driving factor behind him running away in the first place. And because he runs away, he puts Jem into the position where he needs to be the adult and tell Atticus thus earning him Scout's scorn.