How do Airbus and Boeing aircraft handle Dutch roll?
This is quite similar to the other question you asked.

From my knowledge as a flight simulator enthusiast I can summarize the following:
Boeing aircraft in the past used to be controlled pretty much direct pilot with cables running through the aircraft to control hydraulic actuator valves that power the flight control surfaces. The rudder is usually augmented with a yaw damper system which measures the yaw rate and roll rate, the rudder input and aileron input can “calculates” the desired rudder deflection to dampen the yaw rate and dutch roll. This can be as simple as performing a weighted sum on the yaw rate, roll rate, rudder input and aileron inputs with some factors that may depend on airspeed. The yaw damper(s) can be turned off with a switch in the overhead panel and are relatively simple in design. Easy to test fully and certify easily.
When the autopilot is on it manipulates the flight controls directly, i.e. moves the yoke with mechanical actuators. Thus, the yaw damper doesn't even have to know about the autopilot state in this case.

On Airbus planes and more modern fly-by-wire Boeing aircraft the flight control computers are fed with the pilot inputs and the computers then send signals to the actuators electronically by wire. The controls laws within these computers can be quite complex and usually also incorporate the autopilot control modes.

With full control over all flight surfaces the computers can manipulate individual control surfaces perfectly to remove any undesired motions. E.g. in the A320 the aileron and elevator computers constantly evaluate the current pitch and roll rate against the target rates. Any deviations (e.g. caused by a yaw rate) are immediately countered by aileron, spoiler or elevator inputs. With that alone the Dutch roll already can no longer exist. The roll is already stopped because of the roll feedback loop. 

But it doesn't stop there. The flight augmentation computers receive information about the deflections of ailerons and spoilers for example and then can perform internal flight model computations to figure out how much rudder they need to compensate the adverse yaw for example. Or how much aileron will be needed when to stop the Dutch roll as we add rudder deflection. The computers also take the desired turn rate into account and also have inputs such as the balance indicator to fully coordinate the turn to create a slip free turn with close to no side forces on passengers and pretty much zero Dutch roll.

The flight control systems can be almost arbitrarily complex as you add more and more features into it. E.g. when you have an engine failure the computers could also receive inputs from the engine N1 differences and compute the necessary rudder deflection to prevent any yaw from asymmetric thrust. The A380 even goes as far as to use split ailerons on the opposing side of an engine failure to create artificial drag on one side to increase the counter torque. But it can also use ailerons strategically to dampen any oscillations within the wing to create a smoother ride experience. Now you're not only controlling a rigid aircraft, you start to take wing flex into account.

To answer the second question: The yaw damper can be a separate system with parts like asymmetric thrust compensation or other higher level functions just being turned off in case of a computer failure. This way the yaw damper can be used in a more basic control law even if there are mayor failures in other systems which helps control the aircraft in abnormal situations.

Since the yaw damping itself is sort of the last instance before the rudder actuator it doesn't need to be an integrated part of the wider system but the higher level functions will need to be computed somewhere within the control computers. Both implementations probably exist.

... sorry for the long post :)