A dry cough is a non productive cough also sometime called a tickly cough,a chesty cough is more likely to produce phlegm and probably sounds rattly or wheezy,whilst the last one you mention sounds like whooping cough! which hopefully it's not! Perhaps it's a smokers cough? If it you that is suffering tali22 ,get some Covonia,it's good stuff whichever category the cough is.
thankfully its a long time since i had the "vomity" one , but i remember a friend in an exam trying to hold his cough which was so violent he had to hold his exam papers down on the table and you could hear him trying to control the vomit threat- isnt whooping cough in kids only?
both can make you vomit - the tickly one can tickle the back of your throat and make you gag, the other is the phlegm is too sticky and hard to get up could make you gag too