Hi Jamie,
For some reason, songs downloaded with LW 5 don't show up in my shared 'LimeWire Tunes' any more (I'll check with whether this is a bug or a deliberate change by the developers).
I don't know how you were able to move files from that 'Shared LimeWire Tunes' playlist before... that list doesn't contain any music files & so you shouldn't have been able to move those files to another playlist or transfer to your iPod (I can't & I don't know anyone else who can)
LW 5 can automatically add files to your iTunes now. In LW go tools > options > downloads. Tick the box to automatically add files to iTunes > click ok. Now when you finish a download, iTunes will open & you'll see the new file being added. Once the files are in your iTunes, you can add them to playlists or transfer them to your iPod.
If your LW stops automatically adding files to iTunes then reinstall LW. Mine did this the other day. But a reinstall fixed the problem for me.
Late edit - try going to LW's tools > options > downloads. Tick the box to automatically share files with the p2p network. Click ok. Restart LW & download a song - does it add to the 'your name's shared LimeWire tunes' playlist now?