That's because you are adding the songs to WMP from the LW download folder. WMP is a playlist player & will play the songs from wherever they were added from.

I used to recommend 'moving' the song files from the LW download folder to 'My Music' ie: your personal account folder -> My Music (C:\Documents and Settings\"your account name"\My Music) OR C:\Music folder, I think the main one is. Then open WMP up & scan that folder for songs. (You can add My Music as a share folder. I share many artist folders out of my iTunes music library folder.)

Deleting files from LW will remove them from the computer. O.o

Sorry if locations are not 100% accurate. I haven't used windows much for a few years.