Baby acne - Coren's face is covered - but I'm thinking it's an allergy/intolerance rather than plain old baby acne. I'm eliminating foods from my diet to see if it helps.
Cradle cap - It's not just babies who get it - children and adults do too, but it's called something else! :LOL It's just not as visible because we have more hair! It's basically when your body doesn't shed the dead skin cells on the scalp. My dd is the queen of cradle cap - she's 3 and has a bad case at the moment (probably because we used horrible blue hairspray on her hair for fun one day and it wouldn't wash out without shampoo - so we used shampoo and she got cradle cap! ... we don't use shampoo or soap in our house!).
One thing we do to aviod skin/scalp problems is to NOT use shampoo or soap. We use baking soda rinse or apple cider vinegar (acv) rinse. DD had HORRIBLE cradle cap as an infant (didn't start until after her first bath - with baby soap) and as soon as we went (sham)poo-free, her cradle cap disappeared.