Interesting question and I'll be curious how others approach it.
For me, I press down the center of the disk with my fingers parallel with the table surface. This will help strengthen it and it also helps get gasses out to the cornicione. Then, for NY style, I do the knuckle drape / steering wheel technique, then an air toss or two. For Neapolitan, I do the Napoli slap, then drape over both hands to finish the stretch and set it down.
When dressing, especially for NY style, which tend to carry heavier loads of ingredients, I get the sauce to within 3/4" from the edge, and I bias the ingredients towards the outside perimeter. In other words, the spacing between toppings will be greater in the middle of the pie. This compensates for the tendency of the toppings to slide toward the center as the crust forms.
Hope that helps!