Context. Most observational studies so far point towards brown dwarfs sharing a similar formation mechanism as the one accepted for low mass stars. However, larger databases and more systematic studies are needed before strong conclusions can be reached. Aims. In this second paper of a series devoted to the study of the spectroscopic properties of the members of the Lambda Orionis Star Forming Region, we study accretion, activity and rotation for a wide set of spectroscopically confirmed members of the central star cluster Collinder 69 to draw 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. Methods. We study 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. Results. We report very large differences in the disk fractions of low mass stars and brown dwarfs (~58%) when compared to higher mass stars (26+4-3%) with 0.6 Msun being the critical mass we find for this dichotomy. As a byproduct, we address the implications of the spatial distribution of disk and diskless members in the formation scenario of the cluster itself. We have used the Halpha emission to discriminate among accreting and non-accreting sources finding that 38+8-7% of sources harboring disks undergo active accretion and that his percentage stays similar in the substellar regime. For those sources we have estimated accretion rates. Finally, regarding rotational velocities, we find a high dispersion in vsin(i) which is even larger among the diskless population.
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