Currently Azure IoT Edge modules do not support C2D messages. It is mentioned in these docs. Instead of C2D messages, you can use a Direct Method to contact a module, the downside is that the device needs to be online for a Direct Method to work.
To answer your second questions, Azure IoT Edge will attempt to run all the modules at the same time. The exception to that rule is that the edgeAgent module will always run first, and the edgeHub module will only run if there are any other modules running.