Yes the K04 is the stock turbo for the MS3. Cool down is highly recommended if you intend to keep your car healthy and running well.
that stinks. I know all subarus and srts as well as many other turbo cars dont require this cool down period. that why you almost never see turbo timers on any of these anymore. Suabru even put it in print that it wasnt necessary to do the cool down period as newer technology has made this possible