A bridge too far to say that gender roles and identities are purely cultural. Each culture does have its own gender roles and identities but I know of no culture that has none.

It seems we are in the process of building a gender fluid culture, quite the experiment given the known human history. Many people know from a VERY young age that they are a boy or a girl, teasing out whether their preference for gender typical interests and behaviors comes before or after their self knowledge of personal gender is difficult.

I think your last question goes to the heart of this.