you did not ask a particular question but i can comment. any person who been coughing for a month needs to be examined to determine the reason. since you are also having nausea/vomiting which may or may not be related this is another reason for you to see a medical provider so you can be properly diagnosed and treated -- two things that cannot be done blindly over the internet.
i have had a cough for about a month now. sometimes produces mucus sometimes dry cough also have had nausea with vomiting and diarrhea