The LimeWire playlist that shows up in iTunes can be played but not copied or burnt. You need to add the actual music files to iTunes. See Birdy's link about how to do. Unless you do that you won't be able to add those songs to your iPod. To get rid of the listings of LW in iTunes, it looks like there's 2 for 2 reasons. 1. Go to LW's Tools>Options>iTunes>Sharing & uncheck the option to share music with iTunes. 2. Undo the option muhctekdano suggested.
Choose one of those options to get rid of the extra listing.