In this paper, we study continuity and Lipschitzian properties of set-valued mappings, focusing on inner-type conditions. We introduce new notions of inner calmness* and, its relaxation, fuzzy inner calmness*. We show that polyhedral maps enjoy inner calmness* and examine (fuzzy) inner calmness* of a multiplier mapping associated with constraint systems in depth. Then we utilize these notions to develop some new rules of generalized differential calculus, mainly for the primal objects (e.g. tangent cones). In particular, we propose an exact chain rule for graphical derivatives. We apply these results to compute the derivatives of the normal cone mapping, essential e.g. for sensitivity analysis of variational inequalities.