The transform orientation after pressing 'g' and 'z' doesn't match 'Normal' transform orientation I've rotated the mesh to a position that no longer matches the global axis orientation but still want to transform my selection at right angles.

I have transformation selection set to 'normal', the z-axis becomes horizontal (which is ok I guess, as long as I can move in that direction.) but when I press 'z' to constrain my movement along that axis it locks it to the traditional z-axis vertical movement. 

I can manually transform respective to the 'normal' transformation mode by grabbing the arrows but I want to input specific transformation values which is impossible with manual dragging.
First, as batFINGER says, the combination GZZ move object in local Z axis.

Second, why you said 


  I want to input specific transformation values which is impossible
  with manual dragging.


Did you tried? Click and hold on the axis, and enter specific value from keyboard while you holding the mouse button.