Not sure about any special requirements for California. There are special kits sold for this application involving chains. I've also seen boxes supported by threaded rod from structural ceiling above (steel).
The strut/rod hanger is fine, but probably this is a good candidate for seismic bracing. These consist of T-bar wires splayed from the attachment point of the box out @ about 45° to the structure above. (notice that I only said probably, it depends on your particular circumstances). The shaking is due to blade imbalance. You can buy a balancing kit @ Home Depot. It consists of adhesive backed weights, and a set of instructions.
The quality of the fan may have something to with it also. I have seen some cheap fans gently vibrate without the blades attached. Probably has to with the balance of the motor and its mounting in the enclosure.