I was always taught that ‘stitches’ (that pain in your right hand side) is not caused by anything moving or jarring in your body, or pressure building up.
I was taught that it’s caused by a simple lack of oxygen. As you run, especially if you are unfit, haven’t warmed up or have just eaten, you respire aerobically (ie. a combination of oxygen and glucose).
When you start running, you’re still using the same breathing pattern you were using when you were doing nothing. So after a short while, your body is not taking in enough oxygen to supply all your hard-working cells. So, to replace the rest of the oxygen, you start to respire anaerobically (ie. just using glucose, not oxygen) as well as aerobically.
A by-product of anaerobic respiration is lactic acid. When this builds up in your muscles, it causes the pain known as a ‘stitch’.
If you keep running with the stitch, eventually you up your breathing pattern to accommodate the more oxygen required, and the stitch goes away. This is the same reason why you have to breathe heavily for a long time even after you’ve finished exercising - you have to replace the oxygen your cells weren’t getting.
I can’t explain why the pain is specifically located in the right or left side of the abdomen, but the explanation that ‘the diaphragm is being pulled down’ just doesn’t sound quite right to me. Perhaps I’m wrong - I’m not by any means an expert in biology.
About mucus production: again I’m not sure, but my guess would be that because you’re breathing much more frequently, the body increases mucus production to trap the dust and germs that are coming in more quickly. We’re all constantly making mucus, though - it doesn’t suddenly start when you exercise.