Hooboy. I'm not even sure what to suggest. I might just go straight for a diluted bleach solution for the grout , and see if that does the trick. I don't know about the mats though.
FWIW, DD (my now-9-year-old) and her older brother, when they were tots, once opened up about 50 bottles of nail polish in various colors, poured it all over themselves, and made snow angels on my bedspread.