[@Damiann47] - All right, good questions, first of all.

The difference is mostly dependent on power, yes, and how deeply bonded and integrated they are with their Aeon. First Class SOLDIERS and Commanders are top of the line due to this, along with their psychical strengths and years in the programme. To compare a Third Class SOLDIER and a Commander would be like.. a infant to an adult? In terms of their status of bonding with an Aeon and how long they've been training. It can also be compared to the rise in traditional military ranking by certain actions and capabilities. 

You're correct about the normal military. Using more advanced technology and tactics, and given a lot more political tug and pull in regards to power and influence.

Average citizens can be either selected by military officials or the Govern, like drafted. In terms of volunteering, the plausible way to do this is joining the military and selecting for "further training" - which is a glamoured rumour that the individual in question will be "promoted" - the public is unaware of SOLDIERS at this time.