We consider optimization-related problems, from parametric quasiequilibrium and quasioptimization problems to parametric equilibrium problems with equilibrium constraints and finally optimization problems with equilibrium constraints. We propose a relaxed sequential level closedness and use it together with sequential pseudocontinuity assumptions to establish sufficient conditions for parametric well-posedness and well-posedness without semicontinuity assumptions. These conditions are new even for problems in R1 and R2 but we try to prove them in general settings. For topological settings we use sensitivity analysis while for problems on metric spaces we argue on diameters and Kuratowski's and Hausdorff's measures of noncompactness of approximate solution sets. Besides some new results we also improve or generalize several recent ones in the literature.