3 yr old asthma eczema hair loss

My step daughter is 3 yrs old, she has eczema especially on her wrists back of elbows/knees and ankles. She at times also has asthma attacks and has to be given a nebulizer. her assthma is worse when she has a cold and she frequently has colds. She also has an extremely poor apetite and at times it seems like she eats nothing at all, and she will say she is full after only a few bites, her bm's seem normal also extremely picky eater. Additionally her hair is falling out especally behind the ears and her hair is thin throughout. She is always complaining of being cold, but she is also very thin/underweight. She does not like 'rough' playing and gets hhurt easily, she is also very emotionally sensitive. I was wondering if there is a link between the asthma and eczema and also the lack of apetite, and what I can do to help her. She spends about 4 days a week with me/her dad and the rest with her mother.