Below are the answers.
1. I installed the VSE 8.7i client using a specific userid with Administrative rights when logged in remotely via Remote Desktop Protocol. Does this have any bearing on the Auto Update functionality?
No, it will not loose the Auto Update functionality.
2. SHOULD Auto Update work when there isn't ANYONE logged in to the machine? Should it work if someone is logged in but their userid is DIFFERENT than the userid which installed the 8.7i client?
Yes, by default, it will work with System account.
3. How can I debug this?
You need to look into Agent, Update, Mcscript, NaPrdMgr.log.