In the great AutostarX program, what date do I set? Local time or GMT? If I am flying 10pm local time but zulu is the next day, do I set the date in AutostarX to the GMT date?

It really doesn't matter.

The only thing AutoStarsX does is change the planet's positions in the sky...nothing you would really notice in the sim.  The stars still rise and set at the same times according to the seasons and date in FSX.

 

So, you can choose one or the other...it won't make that big a difference.  I personally use GMT.