It ended up being the alternator. Remanufactured Bosch from PepBoys. Worked for only 5 months...

Door lights and "Parking Brake" light were most likely coming up "on purpose" not because of voltage drops because they lit steadily and every single time when battery light was on. They're also red, so this might be Honda's way to attract attention to the panel.

What was lighting up intermittently, especially when battery was going deeper and deeper discharged, was ABS light. On top of it, transmission started slipping and radio going out spontaneously.

I made a mistake in the original post measuring battery's voltage. Indeed, when lights were out, it was about 14.1V... But when the lights were coming on, the voltage was about 12.7-12.7, and even after throttling engine beyond 1500rpm (which was always making lights go out), the voltage still stayed in that low range. I believe despite this, the batter still was _somewhat_ being charged, otherwise I wouldn't be able to drive like this for almost 2 weeks.

Anyway, yesterday engine stalled, wouldn't start again, I had to replace battery, then drive home (new battery didn't help the lights), replace the alternator, and now all looks fine including transmission.

I'll see if PepBoys will accept the broken Bosch back under warranty...