What does Electra being a woman have to do with Sophocles's treatment of her character? If she had been a man, could she have carried out the revenge herself?

Gender roles are given specific prominence by Sophocles from the moment he decides to call his treatment of the Oresteia story not after the man, Orestes, but after Electra.

Chrysothemis specifically challenges Electra in their final argument that she cannot even consider killing Aegisthus herself, as she is a woman and not a man. Indeed, throughout the play, Sophocles explores the idea of Electra as a woman with a man's heart and a man's fury: like her mother before her, she refuses to behave in the way society expects a woman to behave.

Sophocles explores our expectation of men and women, and interrogates the nature of both roles. Why should a man be allowed to do something that a woman is not allowed to do? As to your last query, I think revenge is much easier carried out by a man but revenge by the hand of a woman makes for a better story!