We consider optimization and optimal control problems subject to free and moving boundary nonlinear fluid-elasticity interactions. As the coupled fluid-structure state is the solution of a system of partial differential equations that are coupled through continuity relations on velocities and normal stress tensors, defined on the free and moving interface, the investigation (existence of optimal controls, sensitivity equations, necessary optimality conditions, etc.) is heavily dependent on the geometry of the problem, and falls into moving shape analysis framework.