It sounds like you need to visit your primary care physician
to have an exam, particularly of your ear. Form the symptoms you describe, it is possible that you did indeed rupture your ear drum. The ear drum (or tympanic membrane) is a thin membranous structure that separates your outer ear from your middle ear and the delicate bones and structures that help you hear. When there are sudden changes in pressure in the compartments in front of and behind the ear drum--for example, deep sea diving, flying, or with explosive coughing or sneezing--the pressure differences can be enough to rupture the ear drum. This type of injury
can usually heal on its own, but it will be important to have a physician thoroughly examine your ear to make sure that you don't need a refer to an ear nose and throat specialist
(ENT). In rare cases some sort of surgical or procedural intervention may be required. Hopefully in your case there will be no further action required, but you don't want to take any chances with your ear. The fact that you were coughing and had this happen could also suggest that you had a pre-existing ear infection that might require antibiotics. The bottom line is that you need a physician to examine your ear right away!