We use a large sample of GRB afterglow and prompt-emission data (adding further GRB afterglow observations in this work) to compare the optical afterglows (or the lack thereof) of Type I GRBs with those of Type II GRBs. In comparison to the afterglows of Type II GRBs, we find that those of Type I GRBs have a lower average luminosity and show an intrinsic spread of luminosities at least as wide. From late and deep upper limits on the optical transients, we establish limits on the maximum optical luminosity of any associated supernova, confirming older works and adding new results. We use deep upper limits on Type I GRB optical afterglows to constrain the parameter space of possible mini-SN emission associated with a compact-object merger. Using the prompt emission data, we search for correlations between the parameters of the prompt emission and the late optical afterglow luminosities. We find tentative correlations between the bolometric isotropic energy release and the optical afterglow luminosity at a fixed time after trigger (positive), and between the host offset and the luminosity (negative), but no significant correlation between the isotropic energy release and the duration of the GRBs. We also discuss three anomalous GRBs, GRB 060505, GRB 060614, and GRB 060121, in the light of their optical afterglow luminosities. (Abridged)
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