Maybe the mp3s that won't play are junk... it sounds to me like they might be. Do the file sizes look ok? Try doing a Bitzi lookup on the files & see if anyone else has rated them as useless (search for the file name again & when you see a file with a checked paper icon, that's one that you've downloaded).
How to use Bitzi

See if you can download a different version of the files. Try using the 'All Types' search & look for artist or album, rather than song title. Go for a file with 10 sources or less, or even a single modem source.

That LW playlist that you see in iTunes is completely independant of your library (if that's what you mean). That playlist only shows in iTunes if you have iTunes sharing on within your LW options (tools > options > iTunes). If you have iTunes sharing unticked in LW's options, you won't see that 'your name's limewire tunes' list in iTunes.

If you mean that you want to make a new playlist within iTunes - files have to be added to iTunes library first... only then can you make a playlist.

It's probably just as easy for you to have the files showing in your LW library (I do). They have to be somewhere on your computer, so your saved or shared folder is as good a place as any
As long as you don't delete the files from LW's library then they sould never disappear. But just remember - there's always the chance that someone else might use your computer one day & accidentally delete things. Or one day you might end up with some virus & lose files.

Always back up your files, just in case