Question

A hereditary disease in which the primary symptoms are breathing difficulties and respiratory infections?

Answer 1

Here's a list of inherited pulmonary disorders (also known as diseases):
Asthma, COPD, cystic fibrosis, primary ciliary dyskinesia, pulmonary fibrosis, Birt–Hogg–Dubé syndrome, tuberous sclerosis and hereditary haemorrhagic telangiectasia.

Explanation:

Such conditions include simple monogenic disorders such as Kartagener syndrome and α1‐antitrypsin, wherein mutations of critical genes are sufficient to induce well‐defined disease phenotypes. However, many are complex genetic traits in which inheritance subtly affects pathogenesis, for example asthma and idiopathic pulmonary fibrosis.

Citation: Marinicak, Stefan J, and Lomas, David A(Aug 2017) Inherited Pulmonary Disorders. In: eLS. John Wiley & Sons Ltd, Chichester. http://www.els.net [doi: 10.1002/9780470015902.a0005517.pub3]

Of the above, I assume that you are asking for the more commonly diagnosed hereditary diseases, including:
COPD, cystic fibrosis and pulmonary fibrosis.

The above diseases all cause the symptoms you mentioned, that is, breathing difficulties and respiratory infections. I hope this helps.