They are not supposed to. Liability. What if Mickey dropped your baby?
Funny story (don't let kids who believe in magic read this!) we went with my 2.5 month old a few years back & Mickey was obviously a girl & obviously had a baby my baby's exact age. I say this because of her/his hand motions & communication. I think it may have even been her first day back after maternity leave. Don't worry, we were the only ones there (adults) and our tiny baby (who wouldn't know.) Anyway, she INSISTED on holding the baby for the pic. At first the handler freaked but then I think she had mercy & could see we were Ok with it & it was safe. It is the most adorable photo (not digital or I would post.) And it is a cherished memory of ours.
Also, we did the Disney Visa thing this past trip & Minnie, who wasn't allowed to hold the baby, insisted we all sat down on the floor so she could have our 6 month old sit next to/on her without breaking rules.
Characters adore the kids so they will try to have as much interaction as possible, but really should not hold the babies. And an adult will have to be in the photo. You should be able to position yourself slightly behind the character with the baby in front of you (if you are worried about body shots.) And as far as your face, truth be told I would not wear make-up at Disney if it were not for having to hold the baby for all those photos. But as long as I have a baby, I'll be in full make-up.