As long as you can breathe, the baby will be fine. If you are coughing that hard you may pull a muscle or it might cause pain in your ribs. Both are normal, but don't take anything without calling your doctor. Congrats BTW!!
No coughing will not hurt your baby. Just take some cough syrup. Check with your dr to see whats ok but you will be fine. I coulgh now at 30 weeks and it just hurts cause I have no room but still safe for the baby.