OK, this is a hard one to answer, not only because it's so broad, but because you have to put yourself in a totally different mindset and forget everything you know about biology and psychology. For the record, my dissertation has to do with erotic and homoerotic humor in Renaissance Florence, so pretty close to what you're asking for.

First thing to keep in mind is "humoral theory." This has nothing to do with comedy--it's the idea that human health, personality, and temperament are governed by the balance of the four primary fluids of the body: blood, phlegm, black bile, and yellow bile (this theory survives into the modern day in the adjectives "sanguine," "phlegmatic," "melancholic," and "vitriolic.") Each of these fluids was characterized by being either hot or cold, and either moist or dry. While this theory is far too vast to cover now, the most relevant concept is that of heat. Ancient and Medieval doctors believed that men were inherently "hotter" than women. This gave rise to a number of theories regarding the formation of the fetus. Aristotle and his followers believed the right side of the body to be the hotter side, so eggs or sperm from that side of the body were believed to contain more masculine essence. Some doctors (not all) suggested that a sperm from the father's right testicle and an egg from the mother's right ovary would produce a very masculine child, while a left-sperm, left-ovary baby would be feminine. Some also argued that a left-sperm, right-ovary baby, or a right-sperm, left-ovary baby, would be an effeminate man or a masculine woman, depending on who you ask. So, to a certain extent, some people believed that your gendered characteristics could be innate and determined by your birth.

The more important thing, however, was the "social construction," as you put it, or perhaps more accurately the "performance" of gender. In the Middle Ages it was all about how you behaved. It wouldn't make sense for a man to say "I identify as a woman" if he doesn't actually do anything that's considered effeminate. This behavioral aspect is strongest in the field of sexuality. Sex was conceptualized not as something that two people do to each other, but that one person does to another, through an act of penetration. The penetrative role was masculine and the receptive role was feminine--anyone who violated these roles was upsetting the gender dynamic. The most recognizable effect of this conceptualization was that the old joke, "you're only gay if you receive" held true in the Middle Ages. A man who fucks boys may be labeled a sodomite (also note that in most places, "sodomite" only applied to the penetrator) and prosecuted, but he would not be labeled "effeminate." On the other hand, boys who "let themselves be fucked by men" were referred to by a litany of offensive terms for female prostitutes. A medieval man would never say, "You like to fuck boys, so that makes you less of a man." You were only less of a man if you were fucked by men.

I should also note that in many places, especially Florence, male homosexuality was something of an open secret, and one of those things that everyone tacitly condones despite never saying it in public, especially when it involved young boys. Sodomy was generally accepted as something that boys would experiment with in their youth (before they had socially and publicly established their manhood), but were expected to give up when they got married. This shows in the fact that despite the severe penalties for the crime of sodomy, the rates of arrest and conviction were astoundingly low--the majority of cases are ones that either violate this expectation (i.e. an older married man who let himself be penetrated), or were violent crimes such as rape. I cannot, however, think of any examples of a man wanting to live as a woman or claiming that he feels he is, at heart, a woman. Critics of sodomy may refer to boys being "kept as women" or "like wives," but there is not a single extant written example of a boy in such a relationship referring to himself that way.

As far as women, we have much less to go on, because they had much less access to the literate world and therefore have not left us much documentation from their point of view. A couple of interesting notes, however. We have no idea of the prevalence of lesbian relationships because, as I said, sex was considered to be a penetrative act, and therefore much of what two women do in bed together would not be considered sex. Most historians now assume that lesbian relationships were going on all the time without men's notice. Some argue that the widespread homosexual activity by both sexes was the result of a strictly segregated society. Young girls were kept carefully guarded from men in order to defend their and their families' honor, but girls and boys had plenty of opportunity to socialize and explore their sexuality with their peers. The only time lesbian relationships come to the attention of the authorities is when it involves a dildo or other penetrative instrument, because it was usurping the masculine sexual role. For instance, in Renaissance Florence, a group of women were put on trial for regularly using dildos on each other--in their defense, they claimed it was so they wouldn't be tempted by other men while their husbands were away.