I will start by saying I have to see what you are seing to determine if it is or is not a violation. However, you can do a really good hesitation move without moving your hand under the ball. Essentially, it is an optical illusion where you are cushioning the rise of the ball and making it appear to stop when it really cannot without you either gripping it from above or putting your hand under it (as you point out). It is similar to the illusion of hang time, which is also a physcial impossibility. People contort their bodies in such a way as to appear to hang in the air, but we know they rise and fall acoording to laws of gravity.
As for the spin move, it always starts with the dribbling hand (say the right for this case) pushing in a reverse direction with a simultaneous reverse pivot. I teach this move, and when done properly, there is absolutely no carry involved. The players that carry the most are those who don't get the good push with the right hand and change to the left hand behind the body. The left hand almost always carries the ball forward in this case, because the left hand is not properly postioned for the forward push and the body is already.
As for the length of the movement, I teach my players to send the ball about 10-15 feet on what looks like a bounce pass initially to overemphasize the amount of movement you can legally get out of the initial reverse push by the right hand and to show the direction of ball movement I expect (about 45 degrees left and forward). Obviously they can't continue their dribble, but that isn't the point of this demonstration
But it makes the point that you can redirect the ball to the other side of your body and forward with one good (and legal) push of the ball. 4-8 feet is what I would want out of the move so that the player doesn't stop during the spin move. This amount of movement should be a dribble, not a carry of the ball. If they keep the ball in the air for the 4-8 feet, it would be a carry.