I am wondering if his cough could be caused by a too dry environment? Due to the use of air conditioning.

Yes, a dry environment can potentially cause coughing, especially in young children like a 2-year-old. It is important to monitor the temperature and humidity levels in the room where your child spends time, especially if air conditioning is being used. Dressing your child in long sleeves and pants, avoiding direct exposure to drafts, and watching for signs of overheating can help alleviate coughing and other symptoms related to a dry environment.