Does susceptibility to lung infections run in the family?

Probably not. Unless there is a hereditary immunocomprimised disease, then probably not.
It may but... Transmission is easier in a household due to closer contact and is likely more important than genetic susceptibility in most instances.

Related Questions

Will most lung infections just run its course, or is medicine necessary?

Depends. Some lung infections can run a 5-7 day course and resolve on their own (these are usually viral); more severe infections may require a course of antibiotics. Bronchitis is usually treated for 5-7 days and pneumonia is usually treated for 5-10 days.
Pneumonia. Pneumonia remains one of the biggest killers worldwide. It is the most common reason for infectious related death in americans over 65 years old. It can be caused by viruses like influenza, bacteria like pneumococcus, fungus and parasitic disease. Medications are necessary if the condition is diagnosed correctly. Not treating a known case is inappropriate.
Depends. Infection can be from virus, bacteria, fungus, to name some. So viral infection usually will tun it's course, bacterial and fungal infections will need treatments provided by your doctor after pe and possible diagnostic testings.