So does the user sprinkle the powder after each slice, or does the machine somehow do it? How would the machine make an even layer of powder?
I can't make it up from the pictures, but most printers just trow a pile of powder on top of the last layer and flatten this with an arm. But I think this video will show it much more clear: http://www.youtube.com/watch?v=wD9-QEo-qDk (take a look from 1:50).
Why doesn't the powder stick to the print head, or get pushed around unevenly by the print head?
Because the print head never touches the powder. Just like it doesn't touch the paper in your inkjet printer.
Why doesn't the powder fall out during printing?
Check the video ;)