OK, here's the deal so far.
I serached the archive for "alarm itself" and came up with a mere 18 pages of hits.
The "honking" aspect of the alarm is controlled by a few simple things. There are at least 6 "door switches" and the computer itself. Basically, the alarm will only go off if the comptuer detects that a door has been opened.
On my Explorer, the alarm would go off. I'd click the "unlock" button on the FOB, and there would be silence. I'd click lock, and the alarm would arm. It would be silent for between 2 minutes and two hours. During this time, the interior dome light was not on.
This tells me that 5 of the 6 door switches are OK. Assuming that the computer is not the probem.
That leaves the last door switch. What door, you ask? It's the freaking HOOD! There is a pushbutton switch that is pressed closed when the hood is closed. My reading through the archives here indicates that this switch can fail prematurely. This would set off the alarm.
It looks like an inexpensive part to replace, but you can also just remove the two wires from the bottom of the switch and strip them and twist them together, which defeats the switch and makes the alarm think the hood is always closed. If this solves the problem, then the hood switch is the culprit.
I just sprayed some contact cleaner in the switch now, and I'll see if this solves it. If not, I'll connect the wires together. If that works, I'll get a new switch.
Finally, the red "panic" button on the fob will disarm the honking (until you drive it again) if you press it WHILE THE ALARM IS HONKING. That should get you some quiet for the nigt.
Well, I hope this is all accurate, and it helps. If I find out more, I'll post back.
-Warr