Wow--lots of good questions for a first-time poster. Welcome to EpicSki, Snownat!
To a degree, these questions have already been addressed in this and other recent threads, but let's take another look at them, since they really address the gist of the discussion.
I'll start with your last question--"Is it mostly the elimination of any additional move to initiate (unless you want to change your type of turn)?"
Well, yes--but it's important to remember that what you've described is the outcome--not the cause. Simply "eliminating additional moves" won't help at all if those moves are needed--for any reason. And for most skiers, most of the time, some kind of "additional moves" are
needed to start turns, because they have not moved accurately through the finish of the previous turn--they have not "finished neutral." The "elimination of additional moves" to initiate is the measure of success, but the key to achieving it is to focus on how you finish the previous turn. And quite often, the real culprit is a misunderstanding of exactly what the finish of the turn involves--what does it look like, what are the principles of what I call "neutral" (while others may prefer a different term)--and what are its characteristics?"Can you 'flow' as effortlessly with a cross over, with a cross through, with a cross under..."?
If by these terms you mean "extension through the transition" (crossover), "flexion through the transition" (cross-under), or neither (cross through), then please see the replies to TomB above, beginning with my post #11. Essentially, the answer is "yes"--but these three "options" really represent the end and mid-points on a continuum of possible movement options. Whether--and how much and when--to flex or extend (primarily the legs) is critical in every turn transition (and every other moment in a turn), dictated entirely by the direction of the "ideal" momentum you need your body to have when it exits the turn. As I described earlier, it's the same choice you need to make when throwing snowballs at, um, "targets"--you need to determine the trajectory, especially including how high to aim, uniquely with every single throw. What's right for one throw--or turn--will be wrong for another."Can you 'flow' as effortlessly ... if you ski square, if you ski countered
There will be some degree of "counter" (upper body facing a different direction than the feet) in most "flowing" turns, but counter is, again, an outcome, not a cause. It results primarily from the rotation of the femurs (thigh bones) in the hip sockets when the legs turn and tip, and the increased flexion of the inside leg when inclining into a turn (or the uphill leg when traversing across a slope). Being "square"(upper body facing the same direction as the feet) or "rotated" (upper body facing into the turn more than the feet) generally results from some sort of upper-body-based rotary (twisting) movements--as does excessive
countering. Since these movements pretty much only cause the skis to twist to a direction somewhat across their direction of travel (that is, to skid), they almost invariably reduce speed--thus interfering with the "flow" of the center of mass. They also involve effort. So--in general--anything other than "optimal counter" (which will vary situationally) will interfere with "effortless flow."
Finally, "what happens to the flow when you carve?"
Hmm..... I would suggest that "flow" is all about "carve." The more the skis carve, the more the force from the snow is directed to the side--pushing the skier into the arc, changing direction most efficiently with least reduction of speed. The more they skid, the more a component of that force opposes the direction of the skier's travel (see Ghost's post #15 in this thread)--turning him less efficiently, while reducing speed (ie. "flow"). BUT--you'd better be keeping up with your skis when carving, moving continuously and accurately, if you want it all to happen!