Gender is a pure social construct (e.g. man, woman) and sex is more biologically derived social construct (e.g. male is xy, female is xx). But this doesn't cover all of the known sex chromosome phenotypes, hence there's still a strong social component here too.
I think that depends on your view of sex and what's more important about it. If you view reproduction as the most important (or foundational I suppose, I'm not a philosopher I don't know what this stuff means) aspect of what sex is, there would be two sexes, male and female. If you want an exact representation of all theoretically possible combinations of phenotype and genotype, including everything from surgery to accidents/birth defects and environmental conditions to intentional genetic modification, you are going to have to create a pretty sophisticated "sex" theory (in the scientific sense), but that's not what most people mean.
