Yesterday I learned that the big black rock that muslims worship in Mecca is actually a building called the House of Allah (my Persian girlfriend corrected me once she had stopped laughing) and I'm really curious as to how this house became so central to the Muslim religion.
You're meant to pray in the direction of Mecca and travel on a pilgrimage to this house once a year if you have the means to do so. It seems incredibly important to their religion, but (from what my girl told me) Mohammed didn't use to live there.
Who built this house and why did they do it? I would think that every mosque is technically a house of Allah, so how come this one became so important to their faith?
To put it simply, the Kaaba/House of Allah is considered by the Qu'ran, and thus by the Muslim faithful as the first House of God, or place of worship to Allah. According to the Qu'ran, it was made by Ibrahim and Ishmael on the instructions of God. Because of this, as your girlfriend pointed out, it is the center point to the "Qibla," the point to which the Muslim faithful pray in the direction of. It is also the place to which the "Hajj" and the "Umrah" take places, both of which are pilgrimages to the holy site itself.
As to the historical history to the sight, much like that of the Jewish temple, or any other site with a large and important religous meaning to many people it can be somewhat difficult to ascertain. Mecca, the city in which the Kaaba sits, was a trading hub for hundreds of years before birth of Muhammad and of Islamic history. To what importance is arguable, with some stating that prior to the rise of Islam the Bedouin tribes people and the surrounding societies traded heavily through the spice trade. Others state it was just a common outpost for Bedouin societies to stop over.
What is agreed upon though is that the Kaaba, and the agreed upon holiness of Mecca was not created solely by Islamic belief or teachings. Some form of shrine was present prior to Muhammad and Islam on the whole, possibly to Bedouin idols, possibly to a more monotheistic "God," or possibly to both. This holiness was important, as Muhammad is believed to have helped restore the Kaaba as it was during his life, and when he was forced back to Mecca and Medina, and later conquered them, he reinstated their holiness in Islam. That last part is important to your question mostly because it established the point of the Qibla. Originally it was to Jerusalem, with it changing to Mecca at this point.
So no, Muhammad did not live in the "House of Allah" per se (though Mecca's holiness is related to more than just the Kaaba) it is seen as the first true place of God, and thus you could say the first mosque.