What's the purpose of the folders for every connected client that I see in the Project Window in projects created on Unity?  My projects on local storage does not show this.  I see new folders for EDIT1, EDIT2 etc in the Project Window every time someone works in the project.  We're new to Unity, as you can probably tell.  

Same here. I understand that it has been the default behaviour since MediaNetwork version 4. It likely has to do with the way that Unity manages the shared project, but those folders are generally are empty. If you remove them, new ones will be created for every client that opens a bin or folder.

I renamed by 2 systems ZZAVID1 and ZZAVID2 so that they always appear at the bottom of each list. I do find it mildly annoying, but one gets used to it.

The "first" folder that is created is for the Users personal Project File and Project Settings.  This is how each User can have their own settings.  Sadly, there is a bug too.  So, on some systems,  additional folders get created that are useless and annoying.  Deleting them is fruitless..