We study monotonicity and continuity properties of the p-numerical radius and p-operator norm of n-tuples of operators. It is shown that and are increasing on, complementing the well known decreasing behavior of and We discuss uniform continuity of. Convergence properties of the p-numerical radius and p-Crawford number of converging operator sequences are examined. The results obtained are supplemented by one application. In addition, we develop several inequalities for the p-numerical radius which extend and refine the existing classical numerical radius inequalities.