Is item #1 even a real possibility? After all, the issue is that A doesn't connect, but B does. If there is an IP address issue, shouldn't neither A nor B be able to connect? Also, considering that I shut everything down (including the router) and then started everything back up, wouldn't that resolve any IP issue? Or doesn't it make it very unlikely that this always ends up with A having an issue while B does not?

I don't believe a firewall issue is likely, as I see no reason why it would work fine for a few months, then fail. No changes were made, I was just working on other things. This is a very intermittent problem, but when it happens, it persists for days, apparently. I don't really have enough data to be certain.

As for #5, the services are discoverable, they just can't be connected to via A, though they can be via B.

And for #6, TCP/IP has been enabled from early on. Disabling it does stop both A and B from connecting, but enabling it doesn't fix the problem, since it has long been enabled.

I'm actually leaning towards something strange happening in hardware, but whether the culprit is the router or system A, I can't quite say. Every now and then, when I start A, it doesn't connect to the internet. Rebooting always solves this, and it only happens sporadically. Still, this suggests that there is some kind of intermittent issue between those two pieces of hardware. It seems to me that A sometimes has other issues, even more infrequently, but it's so infrequent that at the moment I can't remember what those issues are/were.