Possibly. While it's possible that you could develop asthma later in life, that situation is less likely than an atypical pneumonia. If the cough is productive or if you're had fever/chills, pneumonia is more likely. It's also possible that you could be developing seasonal allergies that are kicking up with the recent weather change. See you doctor for a possible chest x-ray and allergy evaluation.
Yes & other things. Chronic coughing originates in 4 areas: nose & sinuses (allergy, sinusitis), windpipe (laryngitis, throat cancer, stomach acid reflux), chest (asthma, bronchitis, pneumonia, cancer, tuberculosis), & digestive system (gastroesophageal reflux). Ignoring a chronic symptom like coughing is not a good idea. See your primary care physician for a complete evaluation.