This paper is concerned with the acoustic scattering of a point incident wave by a sound hard obstacle embedded in a two-layered lossy background medium which is separated by an infinite rough surface. Given the point incident wave, the direct scattering problem is to determine the acoustic wave field for the given obstacle and infinite rough surface; the inverse scattering problem is to determine both the obstacle and the infinite rough surface from the reflected and transmitted wave fields measured on two plane surfaces enclosing the structure. For the direct scattering problem, the well-posedness is studied by using the method of boundary integral equations. For the inverse scattering problem, we prove that the obstacle and the infinite rough surface can be uniquely determined by the measured wave fields corresponding to a single point incident wave. To prove the local stability, the domain derivative of the wave field with respect to the change of the shapes of the obstacle and the infinite rough surface is examined. The local stability indicates that the Hausdorff distance of two domains is bounded above by the distance of corresponding wave fields if the two domains are close enough.