I am not sure LW's playlist is copiable to iPod. AFAIK it is not, the playlist is playable but not burnable or copiable. The purpose of the LW playlist is simply for checking & playing your newly downloaded songs. If you wish to get rid of it, go to LW's menu bar, Tools > Options > iTunes, turn off 'Share my Music' option.

You will probably need to add your songs to iTunes music library as described here: How to transfer music to my iPod (& iTunes) {click on link}. Then you can add them to any playlist you add to your iPod. I personally preferred manually updating my iPod. But I don't use an iPod now.