Your mesh object is unnecessarily parented to one of the bones, meaning it is not only being moved by the bones it is weight painted to, but also by the parent, which is why it becomes offset from the armature as it is being moved twice.
Select the mesh and press Alt+P and choose Clear and Keep Transformation to clear the parent.
As for the shaking, you have a dependency cycle, which can be seen by looking in the System Console (the message appears when you toggle in and out of edit mode on the armature):
This means that Blender can't work out how to move the bones properly as they each rely on each other for their position. To fix this you can un-parent the 'launcher', 'pincer' and 'energy pistol' bones. This doesn't seem to effect anything in the armature, so it shouldn't cause any problems.