What are some examples and symbols of the decline of chivalry as well as the decline of the Old South?
(Sorry. I didn't realize until after posting that the question focused specifically on "A Rose for Emily.")
When I’m looking for symbols, I try to keep in mind that a symbol is something specific and concrete that has meaning beyond itself. As a symbol of the decline of the South in literature, for example, I tend to think first of a house collapsing or being consumed by fire.
Faulkner’s novels are always a good source for symbols of the decline of chivalry and the decline of the South. You might want to look at Gowan Stevens’ naïve belief in gentlemanly behavior and chivalry (in the novel Sanctuary), the decline of the Compson family in general (in the novel The Sound and the Fury), or the decline and ultimate destruction-by-fire of the Sutpen plantation (in the novel Absalom, Absalom).
If you don’t have the time and interest to work through one of these challenging novels, you might look at one of Faulkner's short stories, such as “A Rose for Emily.” Kate Chopin’s short story “Desiree’s Baby” might also be worth looking at for similar symbols and other representations of the decline of the South, including the husband’s inability to recognize the reality of racial mixing and the destruction of the mansion by fire at the story’s end.
Yet another literary work that may be full of the sort of thing that you’re looking for is Tennessee Williams’ play A Streetcar Named Desire. Stanley and Mitch represent conflicting male roles, and Blanche is a great example (or parody) of the Southern ideal of the “lady”: she’s disempowered but conniving, etc. It might be worthwhile to list and reflect on all of the references in the play to the family mansion of Blanche and her sister, Stella. At one point in the play, for example, Stanley talks to his wife, saying “I pulled you down off them columns…” In the case of Williams' play, the mansion isn’t burned down, but it is lost along with the rest of that aristocratic Southern family's fortune.