Does a 2004 Mazda3 Htachback have fixed points for roof rack installation?

Yes it does - the fixed mounting points are beneath the black rubber gutter (four of them). The "fixed points" of which you speak are actually 1/4-20 threads (1/4 diameter, 20 threads per inch). The black gutter does come off, though an alternative would be to drill through the gutter once you located the points. I don't know how that would work from experience, though. I believe there are two clips, one at the front of the black trim piece and another near the pillar between the two doors.