Never had a door ajar light problem and thankfully it wasn't one of the gremlins on the list this past week!
I'm glad & relieved to say that the car has been just fine since my last post on Saturday - knock wood! No mystery lights, the battery seems to be holding its own, there have been no airbag warning beeps, and I haven't heard that relay clicking behind the glove box. Tach hasn't twitched either.
I'm not convinced the battery is in perfect shape despite what the garage told me after testing it but I haven't seen any more leakage on top and its voltage isn't bad at about 12.5 volts this evening, a bit higher than the value I obtained this morning. I'll keep measuring it a couple of times a day for the next while.
As for what was going on, who knows really but my money is on the stupid aftermarket anti-theft system being behind all the weirdness.
Without having a full understanding of all the electronic systems & how they work together, what I'd like to think happened is the fan relay went bad and the fan ran all night without our knowing it. Drained the battery way down, and the aftermarket system went into some kind of sustained error state that affected the GEM (or something) when I tried starting the car on Sunday June 26th.
This past Saturday when I turned the aftermarket system on & off (half a dozen times in 2 minutes with a few engine restarts after the battery swap test) it may have reset.
I haven't put the fan relay back in yet - I have had my fill for now and I'm going to leave well enough alone for a few days. Besides I can't find relay in stock at the local shops so I guess I'll have to visit a Ford dealer for a new one. If I put the old one back in and it is indeed bad I want to have a new one at hand to replace it with.