What can cause sudden, moderate to severe right-sided chest pain? It is worse with taking a deep breath and gets worse after pressing on my chest. It hurts along my sternum and in the middle beneath my breast. The left side does not hurt.

R/O PLEURITIS. PLEURITIS=inflammation of the lining of the lung and Pericarditis = inflammation of the lining of the heart can both cause referred shoulder pain. As well as a PE=Lung blood clot. You need a full exam lab testing CXR/EKG/Shoulder X-ray to evaluate your symptoms . Please seek prompt medical attention for diagnosis and treatment options.
Varies . There are a number of potential causes including costochondritis , which is an inflammation that occurs in the part of the rib where the bone meets the cartilage. Pleurisy, pneumonia, are also considerations. An exam and appropriate studies will usually determine the cause. I would not hesitate to be seen.