Look up Cecil’s columns on things like “why does your nose run when you cry?” and the “walking into sunlight makes you sneeze” phenomenon. IIRC, i has to do with there being a whole whack of nerves in the front of your face, so stimulating one bit can inadvertantly trigger a bunch of somewhat unrelated bits.
Don’t know the specifics regarding the sneezing or tearing bit (though I’ve experienced similar reactions pulling a nostril hair). So far as the nose running goes, though, this happens whenever the eyes tear (to a greater or lesser degree). Tears leave the eye by a canal at its medial border and empty into the nose.