Yes. I recommend fragrance free, dye free, sensitive skin products for all family members as well.
Yes. Babies' clothes can be washed with detergent. Parents can use less detergent, and add an extra rinse cycle if they want to be sure the detergent is rinsed out of the clothes. If a baby is having rashes that the parents suspect are from detergent, they can wash clothes with just water for a few weeks and see if the rashes go away; and later try a detergent again.