A sigh is the brain's way of stopping the tiny sacks in your lung from collapsing.
Deep breathing is one of the simplest ways to deal with stress.
Echidnas can survive quite low levels of oxygen.
Hiccups serve no clear purpose. Tadpoles have a hiccup reflex which helps keep their lungs safe while they transition. So our hiccup reflex might be from our amphibian ancestors.
Breathing in through your nose has many medical benefits over mouth breathing. As usual, be wary of misinformation and bias when looking up health on the internet.