Constant burping, occasional regurgitation and hiccups, heartburn and coughing. Do I have acid reflux?

Sure sounds like it. We can't diagnose here, but you've certainly described many features associated with acid reflux (GERD). Usual first measures include dietary and lifestyle modifications and over the counter medication. Seek eval if no improvement to avoid worsening and long term risks. Gastroenterologists are specialists in this area. .