This paper addresses the robust stochastic stability and stabilization of continuous-time Markov jump linear systems (MJLS), with the Markov jump parameters taking values in a countably infinite set. It is assumed that the state and input matrices are subjected to norm-bounded uncertainty with a prespecified structure, which encompasses the block-diagonal setting. We introduce new robust analysis and synthesis characterizations such that, unlike previous approaches in the MJLS literature, the scaling parameters are treated as decision variables in linear matrix inequalities. As a by-product, new contributions to the theory of stability radii of MJLS are provided. When restricted to the finite case, we further introduce new adjoint linear matrix inequality (LMI) characterizations for each of the robust analysis and synthesis problems, as well as for stability radii. Besides the interest in its own right, the adjoint approach allows us to verify that, in the general MJLS case, there is a gap between the complex stability radius and what can be assessed with scaled versions of the small-gain theorem. This suggests a fundamental limitation of the robustness against linear perturbations that the H-infinity control of MJLS may provide. Some numerical examples, which include the robust control of two interconnected oscillators, illustrate the main results.