Before we even get started, let me start by getting a popular myth out of the way. You won’t freeze your lungs from running outside in the winter.
I actually wrote an article about a year ago on winter running and some of the questions I frequently see. If I can get the link right by memory, you should be able to find it here.
As for your questions specifically, the coughing, etc. could have been caused by the dry air. Cold air is dry air. As you inhale it, your body will warm and humidify it. As it passes through your mouth/nose and throat, it is warmed to near your body temperature and humidified. It is actually somewhat normal to finish a run in cold weather with a bit of a raw throat because it is dry but drinking something should solve that problem pretty quickly. If the air seems too cold or dry, many people use ski masks to warm and humidify the air they breathe in before it hits their throats. I do this when it gets real cold and it works great. If you are having lasting problems, you may want to check with your doctor to make sure you don’t have any pre-existing conditions.
Whether to run outdoors or indoors is a personal choice. I choose to run outdoors because I just find getting outside to be one of the joys of running. If you dress for the weather and have no pre-existing medical conditions, you should have no problem running outdoors.