Sex (not sexuality) IS defined biologically. Gender identity or any other social construct of gender is not.
Yes and no; the distinction between sex and gender is overused theoretically, and is a very recent distinction in the first place, one always associated with a certain political or ideological framework.
This is a bit like the inevitable young undergrad who recently heard of the distinction between signifier / signified, and who runs around to "deconstruct" the supposed arbitrariness he suddenly sees everywhere. But to use it that way is a tremendous simplification that doesn't even hold up in the source material of structuralist and poststructuralist linguistics.
Likewise, we can say that sex and gender are separate, and yet the latter is in many ways inextricably caught up with the former, a fact which is not lessened by the possibility for variation within the themes. The amusing thing in this conversation is that the entire existence of transgenderism, as it is self-represented, is a testament to the fact that gender and sex cannot be held apart; for there would be no immense compulsion (under the threat of catastrophic psychological consequences or suicide) to have ones body massively altered by surgery if it weren't for the fact that sex and gender cannot be held in contradiction in a person without consequences. The ability to occupy a gendered position requires that the body be brought into alignment to at least some extent lest it feel inadequate or inauthentic even to the individual him/herself -- this seems to be the underlying logic of the disorder to begin with.