If I set up my Civil 3D drawing in some known coordinate system, e.g. State Plane, in the "drawing settings", can I then toggle my drawing into other known systems, e.g. UTM, by changing the coordinate system under "drawing settings"?
If not, how are the coordinate settings used?
No, You can't toggle them. The coordinate system is set so that if you're brining in data from another coordinate system that Civil recognizes there may be a possibility that the data can be transformed. The coordinate systems are also used when doing a Map Query or attaching FDO. Unfortunately those don't work with Civil Objects.