For a class of quasi-variational inequalities (QVIs) of obstacle type the stability of its solution set and associated optimal control problems are considered. These optimal control problems are nonstandard in the sense that they involve an objective with set-valued arguments. The approach to study the solution stability is based on perturbations of minimal and maximal elements of the solution set of the QVI with respect to monotone perturbations of the forcing term. It is shown that different assumptions are required for studying decreasing and increasing perturbations and that the optimization problem of interest is well-posed.