Great question!
Well the chain is another safe methiod that many vehicle shipping companies use.
Most cars are manufactured with special hooks or holes or spaces in the metal undercarriage of the vehicle so to place these chains because after manufactering, vehicles need to be transported safely to dealerships on such auto carriers. Please note at the end of the chain is a metail hook. Also if you look under your car you will probably notice a hook spot or holes were these chain hooks would attach safely to your vehicle.
You can see a good close up picture of the metal hook on this website http://www.trustandtransit.com/secure-car-shipping.html of chains being hooked for car shipping.
So yes its safe for the vehicle during auto shipping.
Regarding suspension, unless your vehicle has a problem with its shocks/ suspension it will be fine. Vehicles suspension are designed to carry large amounts of weight on them for extended periods of time, this includes trunk cargo as well as people weight.
You can also see an auto transporter using chains to secure a vehicle in the below video:
http://www.karshipping.com/gallery/v/vehicle-shipping-videos/car-shipping-videos/Open_Car_Shipping_Carrier_Securing_vehicle_for_transport.html