How can I burn a playlist to disc if the playlist is on my iPod?
You can't. Songs need to be in a playlist in iTunes in order to burn CDs.
Why can I not add songs to iTunes from my iPod. I drag a song into my iPod, but I can't move it from my iPod to the library - what gives?
Nothing gives. You cannot drag songs from iPod to iTunes, the facility doesn't exist with iTunes software. Never has and probably never will.
In order to get the songs from iPod to iTunes, you can try one of the many 3rd party programs that copy music from the iPod to the computer.
One of the most recommended is
Yamipod. This is a free program that transfers music from iPod back to the computer. However, it does not transfer playcounts/ratings etc.
Other free programs are
Pod Player,
SharePod and
Floola.
If you want to recover just the structure of playlists from the iPod (and not the actual song files themselves), there's
iRepo for Windows. which I understand has this feature along with all the standard features for these programs.
iPodRip also has the feature enabling you to reconstruct playlists.
There is also
CopyTrans. This does preserve ratings/playcounts etc if those are important to you but this program is not free. It also supports video transfer.