Child with severe asthma and allergies. Is it hereditary?

Probable. Asthma is often hereditary as are allergies. However whether a disease condition will develop or not is dependent on how the body interacts with the environment. For example, in identical twins there is a close to 60% that the other twin would develop asthma if one of the twin has it. In non-identical twins, the chance is slightly under 30%.
It is possible. Allergic diseases can "travel" in families however, parents don't pass on specific allergies or specific diseases. For example, a parent would not pass on their cat allergy to their child and if a parent has allergic rhinitis it does not mean the child necessarily will.