“How long have you had that cough?” is something you might ask a friend in casual conversation when you notice him coughing. Doctors, on the other hand, often ask “When did that cough start?”
In either case, while “coughing” is the correct verb, many native speakers (American English) will refer to someone’s cough as something they have rather than something they’re doing, but there are plenty of exceptions.
“How long” and “how long ago” are very common phrases for asking about duration or onset of something. “Over what period of time” is not ungrammatical, but sounds stilted and odd in conversation.
“I have been coughing for over a week” or “I started coughing a few days ago” are typical replies, as this sort of question rarely calls for a precise answer unless asked by a healthcare provider.
The following illustrates this usage in informal, conversational English.
Adam: Hi, Betty. Wow, that’s a nasty cough you have there. How long have you had it?
Betty: Just a few days. I’m feeling better already.
Formal writing or doctor/patient conversations may certainly require more precision and fewer sentence fragments, but “how long” is still likely to be the most common phrase you encounter.