D..
I think that you need to get a second opinion from another doctor about the baby. Coughing certainly can cause vomiting (gag reflex). Discuss the possibility of reactive airway disease with the doctor. It's a precursor to asthma. My son had it around that age, but he did NOT end up with asthma, thank goodness.
Your doctor needs to be giving you some medicine to turn this thing around with his bowel movements. I don't think you're getting enough proactive help and your baby's intestines are suffering. Time to take the bull by the horns here with the doctor.