The vast majority of videos you can download from limewire are illegal, and thus not something we can talk about on these forums. But making the huge assumption that you have downloaded some that are not copyright protected, we might be able to help.
It is probable that the videos are not in dvd-video format. Most video you can download has been converted, or created in other formats like xvid avi or .wmv. Look at the extension at the end of the files you have. It might be .avi, .mp4, .mov, .mkv or .wmv. Not many of those can be read by a stand alone dvd player. Even modern players that have support for different video formats other than dvd-video usually only work with avi files using either the divx or xvid codecs.
If the file you have is a Video_TS folder, with .vob, .ifo and .bup files within it, then you have actual dvd-video.
So the first thing to do is to determine exactly what you have. Then you need to decide if it needs to be converted to something that can play in your stand alone player ( I have no idea what capabilities your particular player has). For conversions to dvd-video look at a couple of free programs called AVStoDVD and DVDFlick. Google those names and you'll find where to download them.
Oh, and the reason you are getting sound but no picture on some computers is the fact that you are missing the necessary video codecs on those machines needed to play back the movie. You should download and use a different media player on those computers...VLC is free, self contained (doesn't require outside codecs) and can play most anything you can throw at it. www.videolan.org