I'll start with the good news:
I unplugged the DSL modem from the A3, and logged into the A3 with the PC. I deleted the rules, and made a new test rule to turn off the lights, set for the time two minutes ahead. It worked.
Since the original rules triggered at sunrise and sunset, I wonder if the A3 was dependent on receiving the times of dawn and dusk from the internet? If so, the rules weren't working because the A3 was never receiving the trigger due to the internet outage. I rebuilt the rules, and if it fails to turn off the lights at dawn, then I'll change the trigger to specific times.
Maybe an idea for an update is to have loss of internet be a trigger? Is there already some way of doing this, maybe through IFTTT?
The bad news:
It did not fix the issue with the Android apps losing connectivity to the A3.
I live in the Upstate region of South Carolina 15 minutes or so away from the nearest incorporated town. The outage appears to be pretty limited, as my neighbor's service is also out, but other people half a mile away are not affected. Obviously, I don't know everybody in the area, and people keep to themselves and enjoy their privacy, so one doesn't simply walk up to other peoples' houses to ask if their internet is on. If they don't know you, it could be hazardous to your health to show up unannounced.
Tomorrow, I'm calling AT&T to give them the "give me one good reason not to cancel my service with you" speech. We've done without broadband internet access a lot over the past three months, so a 3G/4G modem from Verizon (our current wireless provider) should be quite enough to restore basic web access for our PC, and smart home applications.
I updated the firmware when I received the A3 back in November, but it looks like there is a new version out since.