Yes you can get sick from handling chickens and anything that has poo on it. Campylobacter is fairly common in chickens, and though it doesn't harm them, we can get bloody diarrhea. It is also one of the most common types of food poisoning from eating undercooked chicken. I have had baby chicks inside twice, and both times I got a case of it, even though I washed my hands religiously. It runs it's course in a week or so without treatment usually, but it can be serious sometimes if dehydration occurs.
Is the chicken with pasty butt a baby chick? That is usually caused by dehydration which can occur with shipping, or keeping the chicks too warm. Probiotics or some plain yogurt mixed into food with a little water can help.