Is a child who has eczema also more likely to have allergies or asthma?

Yes: Although everybody's different, we speak of the prototypic "allergic march" - eczema as a toddler, allergic rhinitis by ages 4-7, and asthma in early adolescence. Many people will skip one or two of these problems, but in general, young children with eczema will go on to develop allergies and asthma if they are not carefully treated for their underlying allergies.

Yes...: All of these have allergy in common. So, eczema is due to allergy and asthma can also be due to allergy. So, a child with one may also have other manifestations of allergy.

Yes: Having eczema means he is an allergic individual and is predisposed to other allergies. It does not necessarily mean he will have these other allergy tendencies but his risk is greater than the risk of someone in the general population.