My doctor put me on bed rest with my cold, because I was coughing so hard that I pulled a muscle and was in pain for the rest of the pregnancy in the rib. I suggest that you see your obgyn about this, as every cold and every pregnancy are different. Hope this helps, good luck to you and the baby.
No, but it can be stressful to you and your baby. I suggest go see your doctor. He/she can give you a prescription for a cough medicine that has codeine in it that will take the cough right away and it is safe for your baby. Don't take anything else without consulting with your doctor. You can also apply vicks vapor rub to your chest and on the bottom of your feet and wear a warm pair of socks. Take it easy and make sure you drink plenty of fluids and take your prenatals and vitamin c. I always get a nasty cough when I am pregnant that lingers forever it seems like before I have the doctor prescribe anything.
You should definitely go see your doctor. Whatever bug is causing your cough might be much more harmful to the baby than the cough itself.