Is it possible for 5 out of 6 members in the home to develop allergies and asthma symptoms or could it indicate a mold problem?

Asthma in a family. Asthma can run in families and it is an allergic condition and if you have molds it can trigger episodes of asthma, so if you have molds have them taken care of and see if your asthma and other members improves.
Possibly... When many people in an environment develop symptoms like this, the environment needs to be evaluated for possible triggers which includes mold. There are other possibilities as well. If the home is evaluated and no problems are found, then the family members need to be evaluated to determine what allergies they have, if any.