Lung cancer rarely presents with a cough and chest pain - there are usually no symptoms or just a feeling of being winded quickly. If it does present with a cough, the cough is usually persistent (i.e. it doesn't come and go) and it often comes with coughing up some blood.
It is possible for people at all ages to get lung cancer, but it doesn't sound like you have it. What you probably have is something like seasonal allergies, post-nasal drip, acid reflux, you're sleeping with your mouth open, the air is dry, or a combination of those factors. If the cough doesn't go away in another few weeks, schedule another appointment with your doc to get an X-ray, but I promise it will be fine.