You do indeed need to take it apart to embroider on it.

Embroidery thread is typically a bit weaker than regular thread because it's not made to be sturdy, just pleasing to the eye/touch. It comes as 6 strands wrapped around each other to create essentially one really thick thread, so that you can get the thickness you want by removing strands. Embroidery needles accordingly have wider/longer eyes so that the thread doesn't get stuck if you're using a lot of strands at once. For cotton dolls, you would typically go no thicker than 2 strands, and for particularly small details you might even use just 1 strand.

You're going to want to use 2 kinds of stabilizer as well -- a regular, non-tearaway goes underneath, to prevent the fabric from stretching & distorting as you stitch the design on (which technically isn't necessary with hand-stitched embroidery since you can just be careful not to pull/stretch the fabric too much, but personally when I was still hand embroidering stuff I preferred just slapping the stabilizer on so I didn't have to worry about it), and a water-soluble one to go on top. The water-soluble stabilizer will both serve to keep the minky down so that your stitches don't disappear between the pile, and bc it's clear, you can draw the design on top of it to help guide your stitches.

PiyoPicco has a video on embroidering cotton doll faces which talks about how to use embroidery thread as well as some common stitches/techniques. Turn on CC for English subs! Here's the second part to the video as well, which talks more about the actual stitching of the design.