If I understand correctly, the classical Halachic requirements on kosher slaughter only discuss cutting the trachea and esophagus. Is that enough to kill the animal quickly (and painlessly?) What about the blood vessels? Are those usually cut too? Must they be?
It is discussed at length in Siman 20 of Yore De'ah. Essentially the shochet needs to cut through the Trachea, Esophagus, carotid artery and the jugular vein. Both of these shield the esophagus and trachea at the place where the cut is to take place. This would lead to a fairly quick bleed out with the animal becoming nonsensical in a matter of seconds. Siman 24 discusses what happens if the cut wasn't deep enough. Thus only slicing the major blood vessels but not the breathing apparatus.