Inverse elastic medium scattering problems have many applications in science and technology, such as geophysics and seismic wave imaging etc. In this talk, we first report on the uniqueness result to the shape determination of the polyhedral elastic medium scattered as well as its physical boundary parameters by a single far-field measurement, where we utilize the local geometrical characterization of generalized elastic transmission eigenfunctions near a polyhedral corner. Furthermore, we establish a sharp stability estimate of logarithmic type in determining the support of the elastic scatterer, independent of its material content, by a single far-field measurement when the support is a convex polyhedral domain in R^n, n=2,3. A byproduct result of our findings indicates that if an elastic material object possesses a corner on its support, then it scatters every incident wave stably and invisibility phenomenon does not occur. Finally, we consider the time-harmonic elastic wave scattering from a general (possibly anisotropic) inhomogeneous medium with an embedded impenetrable obstacle. We show that the impenetrable obstacle can be effectively approximated by an isotropic elastic medium with a particular choice of material parameters. We derive sharp estimates to rigorously verify such an effective approximation. The proposed effective medium theory readily yields some interesting applications of practical significance to these inverse problems.