I organize my music on my PC in specific folders that I’ve become accustomed to.  On my old MP3 Player I could just recreate these folders. Why can’t I do this on the iPod? It seems it’s all about tags.

You’re right. The iPod is a database driven player, not a directory driven player.  As such, it gets all of it’s information from the id3tags and not from any folder structure. You can’t create folders on the iPod. You need to edit your tags and make sure they are correct.