Does it matter how long the umbilical cord on a new born is attached before it is cut off? Are there pros to having it stay on longer?

Stay longer except. In general it is better to let it stay longer to allow the blood to return to the fetus, except if the baby has heart defect or low apgar score that needs resuscitation.
Every choice has pbs. There may be some benefit in a short delay in cutting the cord. Yet there can also be problems if that is carried too far. Returning blood cells from the placenta does boost their iron stores and delay a natural anemia that comes with rapid growth.However, if too many cells are returned to baby, it can thicken the blood to the point of risk.Kidney damage or stroke is a rare but real risk.
Big debate, one side. There is a vigorous debate going on right now over this very question. Typically, the cord is being cut immediately after delivery. However, there is a growing argument that cutting the cord should be delayed. The reason for this is that the cord continues to pulsate for several minutes after delivery, pumping baby's blood from the placenta back into its body. Early cutting can lead to anemia.