I had to use some reading to see whether I understand the English used here:
Sex and gender are indeed very related to each other, however the difference is not to difficult I think.
Sex: (chiefly with reference to people) sexual activity, including specifically sexual intercourse.
Gender: the state of being male or female (typically used with reference to social and cultural differences rather than biological ones).
Goku is right that three genders is not enough in all cases. When we get back to the topic we see that there are people that find it important to say that they can say about others what gender they have (even when it contradicts the gender the other say it has), there are others that say it is important to respect the other with the gender the other say it has.
It is not because it would be the truth, but because my belief or culture that I think we should respect others.
Going back to original question.
My view of it seems to be the opposite of goku's "If it doesn't effect me it's unimportant so stick to M/F"
in that it is "Who am I to tell other people how to identify"
While I don't understand all the transgender and 3rd gender and so on stuff it doesn't really effect other people.
And the hole form argument. Many now have a prefare not to say option so...
It seems I agree with kdoggroundtwo.
My answers on the three questions:
Is it entirely something defined by society and thus ourselves?
Not entirely, but the most important is how you feel it is.
Should biological factors have anything to do with gender?
The biological factors someone has influences the gender of a person a lot. However everyone should decide for himself/herself about their gender regardless the biological factors.
Should gender exist at all?
It will always exist, but it would be better when society would make it far less important and officials would stop asking about it.
I think it is an important issue. When we could make the gender issue better understandable more freedom would be possible and society could become less authoritarian.