Gender roles have been specific and mostly static since there were human beings. The fact that women can procreate and men can only play a role in that has kept those gender roles fairly rigid over the centuries. When women began to get equal rights, to vote, to work, to be paid the same, etc., then we saw more gender conflict in that area and it spills over into the home. So that is one source of the so called "Battle of the Sexes".
I would like to pick up on a point made in #2. I personally feel that the battle of the sexes that you refer to needs to be considered as a social construct - as something that is created rather than something that just "is" naturally. Gender roles are something that society creates on the whole, and the gender inequalities and disparities that we face which lead to this battle of the sexes are something that our society has created itself and is responsible for.
Gender is a natural dividing line between groups of people. In the home, alliances are often formed based on gender or age, and we set up a natural gender-based competition by creating all-male and all-female teams. We clearly need to do that, but what it creates is a natural "us vs. them" mentality.
There are at least a couple of different ways that you could answer this.
First, you could say that there is a battle of the sexes because men and women are simply and fundmentally different. You can argue that the differences lead to conflict.
Second, you could say that it is simply a social construct. You could argue that there is only a battle of the sexes because our society creates one -- our society sets up the male-female difference as one that is fundamentally salient to all people. Therefore, we look at things from a male vs. female viewpoint rather than dividing ourselves along some other lines.