When I press either Ctrl+F12 or Shift Ctrl+F12 nothing happens
The usual question: do you have BOTH the "Addon Manager" AND the "Couatl powered products" under the FSX "Addons" menu ?
When I call for push back the aircraft moves but no vehicles are seen
What Pushback ? If you are referring to the GSX one, than GSX and its hotkeys ARE woking. If, instead, you mean you called the DEFAULT Pushback, that's it's just normal the aircraft would move without vehicles, that's how the default Pushback works.