Also, I'll add the following. When talking to the Apple helper (was he a genius too?) on the phone last night he seemed to think that there was something in general wrong with my Wifi. In fact, that was his conclusion: there's something wrong with the Wifi adapter and you should go switch it out at an Apple store.
This was related to the fact that we ran a speed test and the results seemed very inconsistent. In addition, his screen-sharing software cut out quite a few times as we were talking.
Now, when it comes to logging in to Messages, it seems to me (and to the Apple man) that the computer isn't even trying to log in. Ok, once you're into the application, if you try to log in in the Preferences menu, you do see the spinning ... thing. But if you try to log in from the initial splash screen (quit the app and reopen), you don't see much action at all. At least that is what's happening on my computer at the moment, and on yesterday's too. Is this a common experience.
This is what led the advisor yesterday to conclude that there's an issue with a connection somewhere, as though there's a firewall (and, of course, we checked to see if it was on; we also went through all the network settings and set up a new network).
So, I'm not going to guess why these things are happening, but I just wanted to add this idea that Messages isn't even trying to log in. This makes me wonder about the idea that there's a problem with the Apple servers, but I don't want to speculate ...