I exposed your link. It didn't work with the img tags because the link is not to the image itself (ending in .jpg, .png, etc).
I assume it's "Folder in Playlist Position" that you're referring to as part of the playlist? That's actually a folder within a folder. Playlist folders don't have an icon unless there's something in them to expand or collapse, and then you get the arrow.
If I right click on the Playlists node and create a playlist, it appears at the very bottom of the list, but then moves to alphabetical order within the folders when I'm done naming it. If I right click on an existing folder and create a playlist, the same thing happens within that folder - starts at bottom (under the playlists), then alphabetical order for folders within the parent folder.
Try opening your Playlists folder in Windows Explorer, and watch what happens when you add a playlist folder within MusicBee. You'll see it created in Explorer, too.
I am observing the same thing you describe with the disappearing folders when dragged to the playlist node, so that's a bug for sure.