I'm going to make people mad, but oh well, here goes. First Cradle cap is NOT dandruff, it is Seborric Dermatitis. I am 41 and have it on my scalp, eyebrow area and creases of my nose. Sometimes it spreds to my cheeks and ears. It is not caused by too much oil. It is caused by overproduction of skin cells and there is NO known cause or cure. I have a cream called Elidel that helps but you cant put it in your hair. It does get worse if you wash it too often and, for me, eating less sweets and fruit and more protein and fresh veggies helps.
Do NOT use vaseline! It is a petrolium product and therefore IS oil! I use SelsumnBlue Max strength (even on my face), but I'm sure it would be too harsh for a baby. Both my daughters had it and both seem to have outgrown it (cross fingers, knock on wood, etc) Mineral oil can help loosen large scaly patches before washing, but oil left on the hair will just make it worse as it clogs pores and looks bad.
I hope this helps, and no, I don't want to hear from the people I ticked off. A pediatric dermatologist (hard to find) could help with both problems although one persons thought that the black spots could be blood, could be true, I have scratched til I bled and scabbed. Yea, I know, yuck, It does suck.