Is gender socially constructed or is it based individually?

It is determined at birth (XX and XY)

Individually.

Whether females can be warriors is definitely determined by culture/society

There are definite biological differences between the sexes which influence our behaviour. Social constructs may encourage these differences, or try to suppress them, but they don't create them.

you are born with your gender

Gender is a biological unit in sexual reproduction. It is a mating strategy that perpetuates a species and produces genetic diversity.

You're referring to gender identity, which is not to be confused with gender. Gender is defined prenatally by chromosomes. XX is female, XY is male, and the rare XXY is intersex.

In regards to gender identity, I doubt it's social. I've heard it said that many people with gender identities that don't conform to their gender have felt so their entire life.