How to calculate damping ratio and natural frequency of a high order system?
Systems that are higher order are composed of smaller poles, so you can find the dominant poles (I'd use a bode plot and find the peaks, if any) to find the natural frequency. As far as I know, there is no damping ratio for higher order systems (it is dependent on the smaller order systems). Overshoot is best found by simulating (with a step input). 

Do a partial faction decomposition and look at the resultant second (complex) or first order systems.