The dome is used to read the light illuminating the scene, not the scene itself. The gray card is used to represent a typical scene, not be a part of one unless you put it in the scene and spot read it. In any case the meter should be calibrated to give the same results, or at least close.
Not necessarily, it depends on the where and how the light falls, and where the gray card is for the reading. Example: wooded area in bright sun. If say, outdoors, in a field, nothing to obstruct or alter the light, then, in theory, should be close to the same.