Hi Doug,
I’ll answer your questions separately:
Q1: For your #5 suggestion, are you saying to remove all of Office 2011 (all apps) and reinstall? Or, just Outlook?
A1: I would suggest you reinstall all of Office 2011, in case there is any conflict caused by the other Apps.
Q2: What program installs that button? If someone had Outlook but did not install Lync, would that button still appear (I'm guessing not)?
A2: installing Lync will install the button, and it will disappear if someone had Outlook but didn’t have Lync installed.
Q3: Do you think their URI is stale and causing button problems?
A3: Is the issue happened before the changing URI thing, or after (right after)?
To solve the issue, I suggest you perform a clean uninstall of Lync for Mac, then install it with the newest updates. For details, please refer to: How to perform a clean uninstall of Lync for Mac 2011.