Ask your childs school what their policy is. Our school is fever over 100 degrees. Coughing and sneezing is part of life and happens daily. If he is truley sick he needs to stay home. If cough medicine like Delsym stops it for a while then dose him and send him. You could just about bet money there will be other kids there with the same, or worse, symptoms.
Our local Head Start called me one day to ask me to go pick up K. She started coughing when she laid down for nap and it was annoying the teacher. I said it would be a while and picked her up just a few minutes early. It's nothat I didn't want to help out but that it was just a teacher who wanted to have her quiet time int he classroom, maybe she wanted a nap too.
Anyway, it's up to you and your school policy. I say give some cough syrup, take cough drops, maybe see if it's excess drainage from allergies and does he need allergy med too.