How does Judge Taylor's cigar relate to the central conflicts? What events occur around the cigar-- before and after it appears?
What an interesting question! Scout describes someone who gets to watch Judge Taylor chew his cigar as "lucky." She talks about the "dead cigar" reappearing as a "flat, slick mess" of tobacco and Taylor's "digestive juices." She also asks Atticus how it is that Mrs. Taylor can stand to kiss Judge Taylor knowing he has such a disgusting habit. Here you get two sort of opposite viewpoints: the first is Scout's childish fascination in "gross" phenomena; the second is Scout's attempt to understand more adult situations and events. To me, this is a great illustration of one of Scout's central conflicts, which is the coming of age/end of childhood innocence.
Before the cigar appears, the court runs very much by the book. Court is called into session, opening arguments are made, guidelines are set for the proceedings. Once the cigar appears, however, the feel of the court becomes much less formal. Scout describes Judge Taylor as being prone to propping his feet up and closing his eyes on various occasions while enjoying his cigar, so we have a similar expectation for this case. The cigar also determine, to a certain extent, the length of the trial. Judge Taylor becomes less patient with witnesses and proceedings as his cigar disappears. By the time the cigar reemerges as the tobacco slick, the trial is winding down.