Using the results of our first paper on the Chandra HRC observation of the Orion Nebula cluster (ONC), we explore the relation between the coronal activity of its 1 Myr-old pre-main-sequence population and stellar parameters. We find that median X-ray luminosities of low-mass stars (M/M 3) increase with increasing mass and decreasing stellar age. Brown dwarfs (0.03 M/M 0.08) follow the same trend with mass. From M ~ 0.1 to M ~ 0.5 M, median L_X/L_bol values increase by about half an order of magnitude and then remain constant at ~10^-3.5 for the mass range from 0.5 to 3.0 M/M. In these same two mass ranges, L_X/L_bol remains roughly constant with age, until it drops by more than 2 orders of magnitudes at the epoch when ~2-4 M stars are expected to become fully radiative. We find a dependence of L_X and L_X/L_bol on circumstellar accretion indicators and suggest three possible hypotheses for its origin. In spite of improved X-ray and rotational data, correlations between activity indicators and rotation remain elusive for these stars, possibly indicating that stars for which rotational periods have been measured have reached some saturation level. Our study of X-ray activity versus stellar mass leads us to propose that the few HRC X-ray sources not associated with any optical/infrared counterpart trace a yet-to-be-discovered stellar population of deeply embedded, relatively massive ONC members.