To control this many LEDs efficiently you need a more complex control. The datasheet for the Allegro A8512 shows how this could be done. It's not a novice project, but guessing from my armchair position I think you could manage this functionality in an amateurish version for under $200 in parts.
To answer your question as to how they control the color between 3200K and 5600K, they do this by having two major groups of LEDs. One group naturally emits 3200K and the other 5600K, and to get the range between is just a matter of blending them. Take a look at the LED selection on a site like Newark, Mouser, or Digikey and you'll find just about any shape/size/color/brightness of LED you can imagine.