If this were true, every person who plays a winter sport at a high level would be dead from pneumonia.
The cough is from bacteria and viruses and he’s probably catching it at school.
If they’re sweating in their coat, it’s not appropriate for the weather. A kid standing still at the bus stop for 15 minutes in the winter in Northern VA can wear a puffy coat but the truth is that those big puffy coats are made for kids in MN, VT or Canada. A kid running around in any temperature above 40 is fine in a fleece, thin primaloft jacket, or sweatshirt.