How is Mrs. Mallard a dead duck or a sitting duck? Does this little joke add anything to the story?

This isn't a joke explicitly made by Chopin in the story, but we could certainly argue that it is implied through Chopin's choice of names for her characters. Her protagonist, Mrs Mallard, has a name which necessarily makes us think of the mallard, a common type of duck. A sitting duck is a colloquial term for a person who is, essentially, open to attack -- someone who has no protection against anything that might happen to them. This is true of Mrs Mallard in a number of ways.
At the beginning of the story, we are told that Mrs Mallard had "heart trouble," with the result that the news of her husband's death is broken to her very carefully. Her relatives are afraid, then, that by breaking this news to her, they may kill her -- she is a sitting duck, vulnerable to attack and unable to protect herself. However, this first attack, as the relatives perceive it, does not have the effect they feared. Louise Mallard is able to weather it, although she holes herself up into a room on her own and goes over the news repeatedly.
The irony in the story, then, comes in the fact that it is not this initial news which kills the sitting duck in the house. Louise Mallard is in her house, in her vulnerable state, believing her husband is dead, when Brently Mallard walks through the door at the end of the story. It is this shock which actually kills her, although her relatives make an attempt to protect her from it. So, while the reader -- and Louise's relatives -- may have felt a sense of relief that the initial "shot" of shock has not damaged Louise's heart and she has survived, this turns out to be ill-placed. In actuality, Louise has simply been placed in an even more vulnerable position, and the reappearance of her husband kills her immediately through disappointment.