Ok this may sound strange but it worked a miracle on my babys terrible cradle cap.
A old wise lady recomended i try using original (yellow) listerine and dabbing it on his cradle cap with cotton ball and rubbing around. then brush. and bathe your child like you usally would.
Within 2 washes my sons cradle cap was gone. i so could not believe it.
I talked to his dermatoligist (has bad exzema also) and she said theres actually been studys done on it, and its harmless. and works most of the time.
For his skin, i would bathe him only 2 time a week. the less the better. And as soon as he gets out of the baath while still kinda damp rub aquaphor lotion on him, it will help his dry patchs. retains moisture.
Also use like cetaphil cleanser, or a dye free non soap wash. it will help not dry out his skin.
Also you could try a bit of over the counter hydrocortizone on his patches if using different soaps and lotions dont help.
Also try washing his laundry in a dye free wash, We use purex free and that helped a bit to.
These are just some things ive learned in the past 2 years with extreme exzema.
Hope that helps you.