Congratulations on your new granddaughter! Premature infants with chronic lung disease can become sicker more easily from respiratory problems such as viral illness (colds, flu, bronchitis) or reactive airway disease (asthma-like symptoms). Although it is possible that she is reacting to allergens from animals, formal testing at the doctor's office is the best way to determine whether she has allergic reactions to common environmental elements (dust mites, grass, animals, foods, etc).
If the baby has been coughing for a week, the parents should probably have her seen by her pediatrician: the cough may or may not allergy related. The parents can discuss with the pediatrician the possibility of allergy testing at that time. A single allergy test by itself does not give the entire picture for a baby's allergy, but it can give additional information for the parents and pediatrician to use in deciding how to best treat the baby for various symptoms.