Alright, now I got proper hairstyles!

Boy, don't I feel stupid. I just learnt how to add eyes and hairstyles to the races by reading this:

"It took me a while to figure out, but you have to open the Eye menu and the Race menu at the same time. You can then drag and drop eye colors into the Race menu."

That's the same way you add most things in the editor, and it is rather obvious now once you know it...