**edited by moderator**
The reason you are having trouble breathing when you run in sub-zero weather is not because you are freezing your lungs. It is because you are freezing the water vapor inside your lungs creating a dry atmosphere. Your lungs are mucous lined organs. That mucous protects the cells of the lungs. When you create the dry atmosphere you are actually killing off lung cells. Hopefully you do not do this on a daily basis because the damage can be permanent. It will depend on your individual lung capacity, how far you run, exposure time, how often you run in sub-zero temps. Wearing a scarf or face-mask (type used by skiiers, mt. climbers, etc.) would help with some prevention. I`m not a runner but have skiied in many sub-zero settings for many years. Best advice, run where it is warmer (like indoor track)or dress for sub-zeroes.
What is likely happening is exercise induced asthma. It is the only thing, I can think people are talking about when they use the term frozen lung
I am an MD and runner. We run in cold, ski etc in cold temps all the time
I do get exercise induced asthma and use scarf etc as well as albuterol inhaler