True or false most pro boxers dont pivot the foot they push off it?

[jms]

The push is what actually moves the hips, the pivot gives you range of motion.

Does the direction of the push matter, should you push towards the hip rotation or up or in the same direction of the punch or does it not matter so long as it gets the hips rotating?

The hip rotation and direction of the punch are the same direction.

But you should push into the floor as scrap was saying, in the direction opposite your back foot, which thus moves your body towards that foot.

You're making the common beginner mistake of looking for perfect, static technique in an imperfect, highly dynamic environment. Take the few examples you've posted, in each case the attacking fighting is moving forward so his feet aren't planted, which changes the mechanics.