I have no problem working with my Limewire Pro, but many songs seem to download properly, are identified as mp3 files, appear both in the Limewire library and also at home/music/iTunes/iTunes Music, but they do not appear anywhere in my iTunes application when I open it and I can't seem to play these, even if I try to open them with Audion or the like.

If Audion cannot play such music files, it's my guess they are fake or deliberately corrupted files that are spammed around the network.
There are a lot of either fake or deliberately corrupt files being spammed onto the network. Catch a look at these: (1) Virus thread: Typical virus spam file sizes - Make a note of them (click on blue link to see) & also (2) mp3 Bit-Rates Versus File Sizes (click on link). Try to stay clear of particular sized files. And, if possible, try to compare bit-rates versus file-size for the same duration of the song.