Does GSX have it's own keyboard keys to open and close the doors?
No, GSX doesn't try to controls doors. You are supposed to operate them just like you did before.
I tried using Shift+E+# and GSX says it's waiting for me to open the doors.
Since you don't say which airplane you use, it's not possible to know for sure if you simply haven't got the command right (which is Shift+E and THEN, after a brief pause, a number. It's not all 3 together), or the airplane you use it's a 3rd party airplane that use a different assignment, possibly with different variables, with its own custom method to open the doors, like a control panel or an FMC command.
3rd party airplanes, if they are not already internally supported by GSX, require to be configured before using them for the first time, including telling to GSX HOW they controls their door.
All of this is fully explained in the GSX manual, pages 30-31
