where is your butt (or hips) during each phase of the turn relative to the rest of your body, and why? What's the angle, and how far inside or outside of the turn do you keep it?
The only place this can be defined is during the instant the skis are released from the snow between the end of one turn and the beginning of the next turn. The hips are directly over the feet then. During the turn the hips must be inside the feet, but the extent of that is totally dependent of the skier's use of angulation, the desired edge angle, the speed of the skier, and the radius of the turn. The skiers center of gravity is above the hips, about in the area of the upper abdomen. For a turn of a given radius and speed the center of gravity will always be on the same line from the feet. On the same radius and speed of turn, if the skier has more angulation for more edge angle the hips will be more inside. If the skier has more inclination the hips will be less inside.
Where is the COG in each phase of the turn? Again, depends on skiing style. An effective carver will have the edge angle increasing all the way through the turn to handle the increasing forces, and the hips must drop farther inside to achieve these angles (no, not forcing the hips in & down, but allowing them to drop). A skidder with flat skis on the snow won't, can't, ski this way. A really, really, really good skier making GS or bigger turns may incline in the first part of the turn keeping the skis on less of an angle for speed, then make a sharp change to angulation for edge angle to carve through a gate at the end of a turn.