The dangly piece at the roof of my mouth is red enflamed and looks like there is a white spot and causes a lot of pain swallowing or coughing?

Pharingitis. It is called Uvula. If You also have tonsillar enlargement and White exudate on them; Then there is Tonsillitis. Cough, sore throat and Inflammation of Uvula/ Pharynx may need Antibiotic. Please see your doctor . Meanwhile Use Throat Lozenges, and Gargle with salt and warm water. Also take Ibuprofen for pain and Qufenasin for Coughing.