wmelvin;
Does the GA mode also command a flight path? If so, can it use elevator trim to accomplish this?
Be careful when reading material which is not in context. The control law being used by AF447 at the time you refer to was not Normal Law. It was Alternate Law No.2 and pitch protections available in Normal Law had been lost.
The elevator NU command was initiated and maintained by the PF and the Trimmable Horizontal Stabilizer moved accordingly to preserve the elevators commanded effectiveness. Thrust was also PF controlled.
Check this summary of
A330/A340 Flight control laws for further info.