Firstly, do not be discouraged. The fact that you are breastfeeding baby that's the best you can do and give your child. Babies will fall sick, breastfeeding does not mean that your child will not fall sick. In fact, it's good for them to be sick, in that way they can build up their immune system as well.
no, don't think it that way. Each time a child is sick, it's a different virus and strain of virus. Their body & our body will need some time to build the antibodies. Have you brought him to see a doctor?
don't doubt yourself. my boy is exclusive bf as well. but he keeps falling sick and ended up in the hospital. it depends on the family medical history too.