2000 Ford Taurus Automatic Headlights Issue

Have been having an issue with my automatic headlights. For a few months, my headlights would randomly turn off while driving, this would occur for just a few seconds. I assumed it was my battery because it had been dying. About 4 months ago, they just stopped working period; however, only when the headlight switch was in the 'Automatic' position (though my dash lights still come on in the 'Automatic' position, just not the headlights and the headlights work perfectly fine in the 'On' position). I then turned around and replaced both the light sensor in the dash and the headlight switch. Neither replacement has fixed the issue. I spoke with a Ford Dealership today and was told this could also be a sign of a bad GEM module. They want $400 for the part, not to mention they want to flash it to my car. So, in closing, I am wondering the following;

1) Could it be the GEM?
2) Doing some searching on Google, I have found a lot of people stating the GEM does not need to be flashed, but I will need to re-sync my remotes to it. Is this also true?
3) If #3 is true, will it affect the remote entry code on my door?

Thank you for your time and expertise in answering my questions.