Is your DVI on the Z2 set up for video/HDCP mode? Without this, you will not get a picture from the LG (or Zenith) DVD player.
For component, you are seeing the "notorious" pixel cropping problem. The player does not fill the full width (or height) of the active image area in the video signal.
You will get this black border using either component or DVI.
The secrets review of this player indicated that there is some cropping of the image on all four sides, though the right side is the worst at 15-16 pixels.