Most kids this age are very picky eaters, but I do agree (from personal experience) that children will definitely eat when they're hungry. Is your god child losing weight? If not, she is probably getting enough nutrients. When my son was two, his pediatrician told me to think about what he's eating on a weekly basis rather than a daily basis. Our job as parents is to make sure that we're offering our kids healthy food choices. As much as we'd like, we can't make them eat anything.
Sometimes it's about a power struggle, at this age. Two year olds learn very quickly that they can control what goes into thier mouth and what doesn't. My grandmother had the same type of issue with my uncle when he was the same age as your god daughter. She said that she would just leave little finger foods out on the table (i.e. pieces of fruit, cheese, whole wheat crackers) throughout the day and he would periodically graze on stuff all day. Your friend may want to try this. It would be nice if her daughter would sit down with the family at each meal, but she may not be ready for this yet. Also, she might want to try something like pediasure shakes to boost her calorie intake.
I would definitely have her check with the pediatrician about the hair falling out. There are various different reasons this could be happening, but she should get some further evaluation. Hope this is helpful!