How does gender shape our identity? What role does society play in this? Use literary theories to argue points (i.e. Marxism).

The relationship of gender and society is complex. First, it is important to understand the difference between "gender" and biological sex; the latter has to do with anatomy, while the former refers to an ever-evolving set of social norms associated with the terms "male" and "female." I think most people (most people in the U.S., at least) derive a significant part of their "identity" (i.e., "the kind of person they think they are") from gender identification.
Your question asks specifically about "literary theories." I think the "literary" part of gender has to do with notion that "gender" is symbolic, in the sense that it is a kind of "text" that people "write" for themselves and one which is open to interpretation. One of the key achievements of gender theory has been to show the arbitrary nature of gender and to explore the relationship between the gendered individual and the society that imposes gender norms on that individual. Judith Butler, for example, argues that gender is "performative," which is to say that gender identities are "performed" by individuals who decide to "act out" certain gender cues (like dress, makeup, behavior, etc.), thereby signalling their gender identities to others. On the other hand, Julia Kristeva, among others, analyzes gender in relation to a kind of "female writing" (ecriture feminine) and finds that the articulation of gender is bound up in the relationship of the individual to language. To crudely summarize: the denotative meaning of words is male; their connotative meaning is female. All of these ideas emerge from the psychoanalytic tradition, beginning with Freud, but especially draw on the thought of Jacques Lacan and his ideas of the Imaginary and the Symbolic.
You also mentioned Marxism. One of Marxism's contributions to understanding identity has been to situate the individual within a class structure. For gender studies, this has developed into the idea of "intersectionality," or the recognition that "identity" is not monolithic but composed of many separate identities, many of which may be oppressed, albeit in different ways. Intersectionality has become a way for people to understand gender within many different contexts, including race, social-economic class, sexual orientation, and so forth.