Does cold air help your lungs feel better?

Yes. Often when people have infections such as colds, flus or even pneumonia it can be difficult to breathe and they can work up a sweat just taking air in and out. For them it can often feel good and refreshing to take in some cold air. However, if they have asthma or bronchitis, the quick change in the temperature can lead to constriction of the bronchi and can make breathing more difficult.
Yes. Only you can really answer that. If it makes you feel better, then sure, it makes you feel better.