I like questions like this. I'm no math expert but I enjoy trying to think through stuff.

I don't think you can get anything but a circle. Since the sphere is the same no matter how you rotate it, just forget about rotations and angles and imagine lowering the plane through the top of the sphere (or left side, or the right side, it doesn't matter which one you imagine because they are all the same). The plane is either tangent, outside, or inside. Those are the only 3 states the plane can be in. However "in" the plane is, the cut is always a circle.