If PO didn't do proper flushes it would not hurt getting the module rebuilt. I would drain/flush/refill EVERY fluid soon.

No, I would check the connector with meter to find the positive pin.

I'm a little confused here. There is the big multi-pin connector which connects to the ABS module, then there is a smaller electrical connector on the top rear of ABS pump.

The connector on the back of the pump is the one you want to test. Test the female side for voltage and then connect the 12volts to the corresponding pin.

Should that female connector have voltage all the time when the car is on, or only when the ABS is activated by the Autologic?

Check it with key on. The pump should run if you are supplying current directly to it.

I get that, but should there be power on the female side of that plug at all times?

That ABS pump should not be running at all times should it? Or only when called for by the computer?

There should be no power to the female side of connector unless module is bad.

The ABS pump does not run all the time. If it does there is a module issue.

Well, I started by removing the ABS module, and what I found was definitely not good. The was about 1/2" of moist white crap at the bottom of the module. There was a good deal of this substance, and it was a bit gritty. So I cleaned it out, and sent it out to get rebuilt. When it returns, and it gets re-coded, I'll see if I still have a problem with the pump. Here's a photo to give you an idea how bad this thing looked inside:

http://i443.photobucket.com/albums/q...psf3bf679b.jpg

I mean, WTF was BMW thinking when the came up with the location for this module? Couldn't they find a spot that was not prone to get road spray?

Looks like you found the problem,