The way I see it, the skullheart is similar in nature to the Witchblade (from the comic/anime of the same). While the gender of the skullheart is unknown and may not even be defined, I view it as a male entity. Similar to the Witchblade, which is also male, it seeks to completing itself via a female vessel. The whole Faustian contract seems to be something added on by the goddess of the trinity in order to instigate the descent of the female vessel into the skullgirl. And it seems like the skullheart gains power for each host it maintains [citation needed] so it is also likely that it is some form of battery that the trinity are working towards charging up. To what end, I don't know.
So as for why it only accepts female vessels. Opposite magnetic fields attract each other, and like fields repel one another. Assuming that the Skullheart is a masculine entity, the skullgirls is the result of the 'marriage' between itself and the host.