One of the fundamental rules of Catholic theology is that Sacraments are only for the living. Therefore, Father Mulcahy could not and would not have given "last rites" to a man that he believed was dead. If it was within a couple minutes of "death", the Sacrament could have been given conditionally, but this would not have been attempted for a man who was apparently "dead" for many hours.
George Wendt is credited in this episode as "Pvt. LaRoche." However, Dr. Winchester refers to him directly as "Pvt. Mosconi" during his examination. However this is not a goof. Winchester was referring to Willie Mosconi, the professional pool player.
During triage, Hawkeye comments, "The way I see it, the Army owes us so many coffee breaks, we should get 1954 off." This line makes sense only if it is presently 1953. (If the Army owes the staff a year off, it wouldn't make sense for them to still have to work all of 1953 before getting that time off.) Yet the war ended in July 1953, three months before Halloween, so it would have to be taking place in October 1952, making the Hawkeye's comment particularly odd.