Arranging things on a tabbed dock (I'm trying on a different computer), I removed some things with the "delete" menu item.  That didn't make the icon go away, but the icon's right-click menu now only has three items, one of which is the "Remove Item".

A few minutes later, I found out the difference:  I dragged the OpenOffice submenu (a folder shortcut) from the start menu, and found it didn't have the fancy flyout.  So I created a new cutom menu and set it to the same folder.  It worked!  Then I deleted the first one.  That made the second one stop working!  Delete removed the underlying folder full of shortcuts!  On the Windows start menu etc., Delete removes the icon only.  Some even have a popup reminding you this and telling you how to uninstall if that's what you really meant.

Who only knows what else got messed up by deleting the original contents of this tab page <sigh>.  I think this is really a misfeature.

Meanwhile... where is the folder-with-birds icon for the OpenOfficeOrg folder/menu?  It appeard by itself originally, but now I can't find it in the icon set when building the "custom menu" the hard way, adding an item for each EXE in the directory.