As it turns out, this had something to do with the fact that I am logged in via a VPN to the vSphere environment using the a PC where the vSphere Client is installed to access it. I installed the vSphere Client on a machine (a VM) within the vSphere environment and was THEN able to install the VUM plug-in and enable it, and also the vCenter Service Status & vCenter Hardware Status were displayed normally as 'Enabled'. So somehow, there is a resolution issue accross the VPN I need to look into since it still shows locally as 'Disabled'.
UPDATE: I tried 2 different PCs connected via VPN to that network and then I changed the DNS for that remote network connection across the VPN to the DNS server address within the vSphere environment and that seemed to do the trick. One local machine with the vSphere Client is still giving me errors about the vCenter Service Status and Hardware Status, but the VUM installed properly and is enabled... and another local PC with the vSphere Client installed seems to be fine with all three plug-ins (VUM, Service Status & Hardware Status). Must be a local issue with that PC.