I need to know whether the position of the CG is higher, lower, or at the same level within the body when a person is standing up as when a person is lying down? Why or why not?
The centre of gravity is the point around which the weighted distribution of mass of the body sums to zero. Its relative placement within the body is independent of the orientation of the whole body; though of course it is affected by redistribution of parts of the body.
Laying down does not in itself affect the location of the CoG within the body, but positioning of the limbs, and shifting of organs, will have some effect.