Most observational studies conducted so far point toward brown dwarfs sharing a similar formation mechanism as the one that is accepted for low-mass stars. However, larger databases and more systematic studies are needed before strong conclusions can be reached. In this second paper of a series devoted to studying the spectroscopic properties of the Lambda Orionis star-forming region members, we study accretion, activity and rotation for a wide set of spectroscopically confirmed members of the central star cluster Collinder 69 to assess analogies and/or differences between the brown-dwarf and stellar populations of this cluster. Moreover, we present comparisons with other star-forming regions of similar and different ages to address environmental effects on our conclusions. We studied prominent photospheric lines to derive rotational velocities and emission lines to distinguish between accretion processes and chromospheric activity. In addition, we include information about disk presence and X-ray emission.