You should try jwzlyrics. It picks up songs changes in iTunes (and Spotify), finds lyrics and displays them in a window.
It's dependent on https://lyrics.wikia.com/ for it's lyrics, and works quite well for me.
Looks like LrcShow may do the trick. There are full usage requirements and instructions on that page, but it seems that it supports live lyric display for lyrics with bracketed timecodes before each word (.kra), before each line (.lrc), or a static display for lyrics with no timecodes (.txt).
Cover version works on iTunes! What you should do is to create a new directory ~/Library/iTunes/iTunes Plug-ins and put the Cover version file in there.
Go to Finder
Press Alt (OPTION) while selecting GO in the menu
Go to Library
Create iTunes directoy if it doesn't exist yet
Go into iTunes directory and create iTunes Plug-ins
Copy Cover Version ...
First you have to add the lyrics to your song in iTunes (tap on the song, then press cmd + I).
Go to your iOS device (since it is shared via iCloud), and then single tap on cover in following screen:
You see the lyrics, you can switch it back to cover by single tap on the lyrics.
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 ...
For the lyrics part, I would recommend Get Lyrical. It can actively tag songs, tag specific songs, or tag a playlist. It gets all of its lyrics from Lyrics Wiki. This is a great little application and I highly recommend it. It has a simple, clean interface that is really easy to use:
It looks like it does the same thing as Harmonic.
Allows you to search for lyrics by song. (Don't think this is what you are looking for)
Canto Pod (Free)
Allows you to search for a song, then add it to your iTunes library. (Don't think you are looking for this either)
LyricSearch and Canto Pod were lyrics ...