Thanks to Zend support, I have the answer which I will share here in case other new users find the same issue.
On a fresh install, Zend Studio (9.0.4) defaults the workspace to the Windows user profile (mine is in C:\Documents and Settings\username) and creates a folder named Zend there when the product starts up.
You need to change this location as soon as possible if you use roaming profiles, to avoid the problems noted in my post above.
You can do this by creating a folder in a new location (I was recommended to create it on a local drive), then in Zend Studio going to the File menu, then Switch workspace, then Other and selecting the new workspace folder you created. Studio will restart. The studio window title will now show the workspace folder that you selected.
I recommend that you now go to your Windows user profile and delete the Zend folder from there (Windows refused to delete mine, but I moved it elsewhere) if your profile is regularly backed up.