Normalization is just "scaling", this would not affect the sign.
It will. When the first triangle is oriented towards the camera, while the second triangle is oriented away from the camera, the projected sum can be positive or negative depending on the length and direction of the normal vectors. When the vectors are not normalized the length of the normal vector will also depend on the size of the triangles, not only on the direction of the vector. It could be that a quad is warped and distorted so much that a small first triangle is oriented perfectly normal to the viewing direction, and the other triangle is large and points slightly away from the camera, so that the sum becomes negative. Therefore the quad would not be drawn although the overall result might be better to display it. Such a case, btw, indicates that it might be better to render on the basis of triangles. Or are your quads always flat?