Yes, the yaw damper has limited authority. It depends on the airplane, but likely the damper can put less than 5% of full throw on the rudder, and usually much less than that. Remember, the damper is designed to reduce the _rate_ of yaw, not necessarily the yaw itself.
The 747-400 has what you describe as T.A.C. only in the Approach mode, when all 3 autopilots are working simultaneously. In normal single-autopilot mode, the autopilot does not control the rudder (though the dual yaw dampers are still operating).