Ok, here's what I've found so far. I still haven't tried the fix suggested above yet. I want to try to isolate when and where the problem occurs of when GSX shuts down and Couatl disappears from the drop down menu first.
It looks like the main culprit is PMDG B777 and only disappears at the 'Boarding' point. If I choose to just re-fuel, no problem. When I then select to 'Board Passengers' it starts and the trucks roll up to the plane then disappear and Couatl disappears from the menu.
I tried it with a few other aircraft and found these results:
PMDG 737NGX - no problems
CS 757 & 767 - no problems
Airbus X Extended - no problems
Level-D 767 - Couatl appears in menu but can not use Ctrl-F12. Nothing happens. (this may be due to a key mapping conflict though with LVLD)
PMDG 747 - Sometimes GSX works ok for re-fueling and disappears when Boarding and Couatl disappears from menu, other times, Couatl appears in menu but Ctrl-F12 does nothing
PMDG 777 - GSX works ok for re-fueling and disappears when Boarding and Couatl disappears from menu
I did enable logging in the Troubleshooting section and checked the %APPDATA%\Virtuali folder. There was an .ERR file in there but it showed last modified on 5/22/12 so I removed it and re-ran FSX for the problem and kept checking to see if a new one was created, but no, a new one is not created after GSX shutsdown and disappears.
For now, I'm just using the Re-fuel and pushback option and not choosing 'Boarding' to prevent it from closing.