We study a nonlinear fluid-structure interaction problem between an incompressible, viscous fluid in 3D and an elastic structure whose Lame elastic parameters, thickness and density are all functions of space allowing jump discontinuities. This problem is motivated by studying the interaction between blood flow and arterial walls treated with vascular prostheses called stents. A stent is a metallic mesh-like tube used to prop the clogged arteries open. The Navier-Stokes equations for an incompressible, viscous fluid are used to model blood flow, and the cylindrical Koiter shell equations with discontinuous coefficients are used to model the elastic properties of arterial walls treated with stents. The fluid and structure are coupled via two coupling conditions evaluated at the moving fluid-structure interface. No assumption on axial symmetry is used in the model. We prove the existence of a weak solution to the underlying nonlinear 3D moving-boundary problem, and design a loosely-coupled partitioned scheme (beta-scheme) for its solution. The numerical scheme was motivated by the main steps in the constructive existence proof. The existence proof shows that the proposed numerical beta-scheme converges to a weak solution of the nonlinear problem. This is the first convergence result for the proposed partitioned beta-scheme. Several numerical examples are presented where different stent configurations are considered. The numerical fluid-structure interaction solutions clearly show that the presence of a stent induces wave reflections in arterial walls, and significant flow disturbances, especially near the proximal site of the stent.