Since I answered the original question, I will try also to answer this one.
We need to reconsider the phrase "social construction and nothing more", or at least to what you take to be the implications of such a description, that somehow what is socially constructed isn't real. One would need a lot of argument to establish that conclusion. Prima facie, socially constituted facts are no less real than biological or anatomical facts; they are just different. Consider, e.g., facts about political and legal authority. Surely these are socially constituted, but I would not suggest you tell a military tribunal that you can't be guilty of disobeying an order from a superior because social facts are unreal. That should answer question (1), I hope.
Similarly, socially constituted facts matter to people every bit as much (and in some cases more) than biological facts. As I pointed out in response to the previous question, the mere fact that gender is a social (not merely anatomical) matter does not imply that people do not experience it as a fundamental part of their identity. Indeed, since humans are "social animals", there is nothing at all surprising about this. How we relate to ourselves is bound up very much with how we relate to the world around us, and that includes the social world. And that, I hope, answers question (2): Whether gender is a social construction just doesn't bear upon the question whether one's gender identity can or should be important to one's sense of self. Indeed, since it is just obvious that gender identity is important to people's sense of self, it's not clear what's left to discuss here.
So, people do have genders, and they can have them even if gender does not "exist in [the] fundamental sense", meaning, I take it: even if gender is not a biological notion. People can have jobs, and friends, and husbands and wives, and stand in relations of authority to one another, too, even though these do not "exist in [the] fundamental sense" either. One ought not get carried away with the language here, even if one does think there is something to be made of this "fundamental sense" language (as I am not entirely sure there is).
I'd love to be able to point you towards more to read, as I do think these are fascinating issues. Unfortunately, I'm no expert. But you might start with the article on Feminist Metaphysics over at the Stanford Encyclopedia. The soon to be out book The Metaphysics of Gender, by Charlotte Witt, will be technical, but it might be something to which you would be sympathetic.