Log of specific incidents
On Jan. 19, 20, and 21 I was able to start the car with the jump starter device (it consistently seems to require two tries), and drive each time for about 30-60 minutes. Each time, the car will not shut off without intentionally stalling it. The battery light flashes when I do that, and the radiator fan remains running for a few minutes.
We suspect that whatever prevents the car (sometimes) from shutting off is closely related to the parasitic drain. This seems to be borne out, in that the car still needs a jump after sitting overnight. It rained the night of the 20th, but I had a cover over the sunroof. Advance Auto Parts will run a test for free of the battery, alternator, and starter. So far I have not taken advantage of this, because the car will not stop properly.
The car-not-turning off could either be en electrical problem or "dieseling" according to this website 
. More likely electric since the engine continues operating smoothly.
Removed the first two panels below steering wheel, Jan. 21. Need to be thoroughly cleaned with Goo Gone. Next step is not immediately apparent as ignition module is not fully exposed. As we knew, the plastic around one of the screws was broken.
Jan 23 borrowed Dave's charger. Unsure of the numbers on the readout, but I charged for a couple hours at 10 amps, and it seemed to go from ~3.9 volt to 13.1 volts (using the jump starter's device to get that reading). Have not tried starting, much rain forecast for tomorrow. Enhattened.
Useful links
This seems to validate that moisture is shorting a component. But it offers no detail. Yahoo Answers
Here is a Golf TDI specific troubleshooting guide. Covers many years, has a wiring diagram...many good resources here TDI Club
Here is an overview of how to find a "parasitic draw" (note the term, useful for searching?) 2carpros.com
General overview at itstillruns.com
A discussion in which the owner of a 2002 Passat (similar car) is seeking a parasitic draw; experts seem unanimous that he should test the battery (which aligns with Ward's suggestion). passatworld.com
Somewhat useful video overview of diagnosing and replacing ignition switch in VW: Humble Mechanic
Can an ignition switch cause a parasitic draw? Trouble loading this site .
Less useful links
I *think* we can rule out the failing coils, distributor cap, etc. issues that are discussed on some sites, because those seem to involve arcing electricity when the ignition is cranked. That's not the case here.
A forum post about a Chevy (focuses on distributor cap and ignition system)
Probably useless chat from cartalk.com
Very useful video for probably a different problem (starter cranks but doesn't turn over): Distributor Cap video
Ward's suggestions
Battery tester. 10-12 dollars. napa
Inspect the fluid levels in the battery. The plates should be visible in every cell and fully covered by fluid. Use a flashlight. Add distilled water if required.
There are many types of integrated battery testers, each one corresponding to a specific condition testing procedure, according to the type of battery being tested. Their common principle is based on the empirical fact that after having applied a given current for a given number of seconds to the battery, the resulting voltage output is related to the battery's overall condition, when compared to a healthy battery's output. wikipedia google
Test the specific gravity of the fluid. This will establish the degree of charge. Driving for 30 minutes should show as substantial charge. Drive further. Test again. Test to see if and how fast the battery discharges when parked.
See:
* Find a Parsitic Drain (WikiHow)
* Another page on how to find a parasitic draw
* • TDI Club