And why does the female bird carry the Y-chromosome?

I don't know, but butterflies are the same way. All humans (and most other animals, I assume) start out in embryotic form as females and then at some point in all of those cell divisions, the male y-chromosome kicks in. The book (Genome, Matt Ridley) says that the sex genes are the most volatile of the chromosomes. The x and y are constantly fighting with each other (seriously). Certain "improvments" in the x are deleterious to the y, and the y has been shedding excess genetic material for a long time. That's why it's so much smaller than the others. With all of this changing going on between the two, it's a fair assumption that it was all switched at some point, or that birds and the ancestors of mammals emerged from the asexual goo on opposite sides of the gender coin. But really, I don't know.