Mouth breathing is thought to be a mild issue for some people. It can (according to some research) increase your heart rate at rest, but I would not think enough to give you chest pain. You should see a doctor about it.
If your doctor rules out heart issues, then it could be anxiety. Indeed some research shows that mouth breathing can increase anxiety. One thing you can do at night is lightly taping your mouth. As long as you don't have some sort of issue breathing through your nose (or other sleep issues), then this can break the night mouth breathing habit.
As for running, most people breathe through their mouths. You can try nose breathing on your easy runs. It could be difficult at first. If so, try taking some brisk walks (with hills). That should ease you into it. I don't think it's a bit deal to mouth breath while running, though.