When I was animating my character for a simple run I notice that when I use the Paste X-Flipped Pose, It moves the hips for some reason and it should be doing that.
I don’t know really. You seem to have some transforms for the armature in object mode, apply object transform location to fix that.
You should probably do something about the eyes too. Why do you have two rigs? You can join the rigs and parent the eye controls to the head and it should work better.