Mourgo,
Another thing to consider is that the 777 is fly-by-wire. It does not actually have a rudder trim as you mean it. Everthing is done by the PFC's and ACE's inputs to the power units. In the photo, the tab you can see deflected in the sane direction as the rudder is a flight tab, not a trim tab. Basically, it's smaller size means that it can be deflected more in to the slipstream for the same hyd power input. Also it is at the back of the rudder, so the moment arm is greater and it has more effect for the same deflection. At higher speeds, the main rudder may not move, only the tab. A trim tab would be deflected the opposite way, so providing a force to pust the rudder.
The 777 does not have a yaw damper as such, this function is carried out by the PFS'c/ACE's. On the ground it relies on internal friction to stop the rudder flopping around and breaking something.
If you look at the 777 on the ground, you will often see the flaperons drooping unequally, same reason, no Hyds., rely on internal friction to hold them up.
Bolty, It must be assumed by the fact that you made no attempt to answer the question that you had no idea either.