1 - Do you have some general idea of what your transfer function looks like? Or at least a frequency response? Could you perform some kind of step response? You could then try to fit an order-2 model on your step response and then try to tune your controller.
2 - You mention that your system is underdamped. I assume that you have underdamped stable poles (...
There are different ways of expressing a control system. For example,
Observable Canonical Form
Controllable Canonical Form
If you express your system in controllable canonical form, then $B$ take the form of a column matrix with all zeros except one in the top position.