Could heartburn damage my esophagus?

Yes. It could make it swollen and irritated and cause long term changes that could lead to precancerous or cancerous lesions.
Yes. Heartburn occurs when acid from the stomach refluxs or splashes up into the esophagus (and sometimes the mouth or throat). Complications from GERDS could include development of esophagitis, or possible a peptic (esophageal) stricture. Barrett's esophagus can sometimes lead to esophageal cancer.