“Do your lungs hurt with bronchitis?”

Rarely

From my experience with multiple patients, you can have pleuritic pain (lung pain) as the membrane around the lungs get inflamed, this typically is more common with pneumonia, but can occur with certain stages of bronchitis. However, often patients with bronchitis develop a persistent cough which can start in the intercostal muscles (muscles around the ribs) causing pain as well, often referred to "lung pain" by patients.
Hope this helps and answers your questions.

If you are coughing your lungs out yes. Take a cough suppressant like dextromethorphan or over the counter cough syrup. If you have a fever or pain persists, see a doctor immediately, it could be pnuemonia.