The requirement is that if your atlas is , then for every and every , you have . It sounds like you might be forgetting that the matrix (and hence its determinant) depends on which point in the chart intersection you look at it from.

Now, since determinant is continuous and we never have determinant zero (these are diffeomorphisms from ), the determinant is either all positive or all negativeon each connected componentof . But of course there might be several components--for example, the usual construction of the Mobius bundle over .

To answer #2 (if I understand it correctly), consider with the atlas given by stereographic projection. There are two charts, their intersection is which is connected for n>1, and the determinant can be +1 or -1 depending on how you define your projection.