Making a plant stable with a compensator based on given requirements
Your intuition to use a compensator, not a PID, is correct in this instance as you need to meet gain and phase margin requirements, which is generally what compensators are for--you can easily get a. and b. using the right compensator. The gain crossover frequency in the Bode/Nyquist plot is roughly the reciprocal of the timeconstant for the response--e., the steady state error can be solved for using any controller by taking the DC limit--f. Overshoot always has to do with damping so you could work out what the effective damping ratio of the CLTF is and constrain it that way--d. Finally, I've never heard of c. 

Another approach which is likely to work with much less effort is to select a compensator, put the poles/zeros where you get a good frequency response for a/b, then tune the poles/zeros around these points, checking the response characteristics using a simulation.