Can someone be in so much emotional pain that they start feeling bad physically?

Yes. I would advise being away from the source of pain eg the internet or a person or thing or thought causing the pain and try studying hard picking up a hobby, spending time with family or friends helping someone or volunteering at a community center or charity . Keeping your self busy. You are young and your possibilities are limitless. You can do it!

Yes. The ideal is to manage the emotional pain. Distraction, like taking a hobby is not a good way to deal with it. It is best that to reduce emotional pain by avoiding triggers. It's easy to say,.. but emotions are an internal expression of your perspective. See a therapist and/or doctor for more information.

It is not feasible to provide a meaningful opinion without taking additional history, physical examination and may be some tests. It would be prudent to see your doctor and discuss if you may be depressed. Counter-intuitive as it may seem, exercise may help. Start with low intensity exercise such as walking and gradually increase the intensity as tolerated. Drink more water. Wish you good health!