Mylenium,
Checking the router log was a good idea. It came up clean... except for discovering that I had forgotten about updating my router firmware a week ago. So, I may try bypassing the router to eliminate it as a cause later today.
I checked out SysInternals Process Monitor. I was pretty overwhelmed with the amount of detail it provided and all of the flexible controls it offers. I didn't really know what I was looking for. So, I poked around with it blindly for about fifteen minutes. I immediately found the App Manager's update process and tracked its activity. I found the location of a log it was keeping. And, one thing in the log stood out. But, other than that, I didn't see how the monitor might help. But, again, I didn't really know what I was looking for. Any advice?
The thing from the updater log that stood out was the way the relevant error message was worded: "network connection was unavailable". Knowing the programming behind processes like these (and having done a lot myself), I understand that it could simply be a message from a higher-level process, leaving out lower-level results which may indicate it is actually a connection to a particular server that was never established. However, as it reads, the updater cannot make any connections whatsoever. Either way, I still do not know how to check into it.
The packet analyzer (Wireshark) seems more like the right tool for the job. But, the learning curve for that sort of thing seems like too much of a time investment. And, having no experience with packet analyzers yet, if I don't drown first, I will probably get incredibly distracted finding out everything it can do.
Your suggestions are solid, though far from the silver bullet I was hoping for.
