How can I view iTunes metadata lyrics without using Get Info in iTunes?
Have you tried Lyritop? It's an app that displays the current iTunes song lyrics on the desktop. It does display a dock icon. You can also adjust font, size, position, color, shadow, etc.

There are no individual lyric files that are stored on your computer. The lyric information is embedded in the id3 tag of each song file. There is an app called MP3LyricsExtractor that will batch extract song lyrics into individual text files.