How was the Catholic Easter calendar established, despite this passage? Matthew 12:40
For just as Jonah was three days and three nights in the belly of the sea monster, so will the Son of Man be three days and three nights in the heart of the earth.

If Jesus said he would be three days and three nights, why are the events of his death and resurrection commemorated on Friday afternoon and Sunday morning, a span of one day and two nights; specifically, how were the memorial services for those events established?
Note that I'm not asking for the teachings on when the events of that week happened, but how the Easter weekend masses were established and why they occur when they do among the Catholic and Orthodox churches, and about the differences among the two.
The Days of the Week 

First, note that according to the Jewish calendar that the "day" begins in the evening, so that the "day" of "Friday" would comprise Thursday night, Friday morning, and Friday afternoon; "Saturday" would comprise what we call Friday night, Saturday morning, and Saturday afternoon; and "Sunday" would comprise Saturday night, Sunday morning, and Sunday afternoon.  This is why, for example, Jewish synagogues hold Sabbath services on Friday evening.  When I was in Israel (in Haifa), I heard horns throughout the city announcing the end of the Sabbath on Saturday night, not Sunday morning.

This still doesn't completely reconcile your question regarding three days and three nights.  Theophylact's commentary on this passage (11th century Byzantine) implies that "three days and three nights" did not imply three complete days and three complete nights, but rather that the event would take place over the course of three (Jewish) "days", as defined above.

The early Church observed the same scheme and the Orthodox Churches continue to do so.  The troparia and kontakia (Byzantine-style hymns) that are sung in commemoration of the saints during the evening Vesper services are for those commemorated on the following calendar day.


 Day on Which Easter is Celebrated 

When I started to research the answer to your question, I thought that it would be fairly straightforward.  It was anything but.

The day on which Easter was celebrated - called Pascha by the early Church (and still - even in English - in the Orthodox Church) was not consistent throughout the Church until the 1st Ecumenical Council at Nicaea came to agreement in 325 AD.  Very early on, it seems that Easter may have been celebrated on the same day as Passover, but at one point this was forbidden by the Apostolic Canons:


  If any bishop, or presbyter, or deacon celebrate the holy day of
  Easter before the vernal equinox with the Jews, let him be deposed
  (Apostolic Canon VII).


Up to the Council, it seems that the Sees of Rome and Alexandria had one formula, while those of Constantinople and Antioch had another.  Constantine I, in a letter to those not present at the Council, wrote:


  By the unanimous judgment of all, it has been decided that the most
  holy festival of Easter should be everywhere celebrated on one and the
  same day


Although the method to be employed seems not to have been explicitly included in the Acts and Canons of the Council, it was documented elsewhere by Eusebius, Chrysostom and others and is described in a deceptively simple formula:


  Pascha [Easter] is the Sunday immediately following the first full
  moon after the vernal equinox.


How the precise date was determined each year in antiquity, however, was a very complicated process, described in detail by James Campbell in an article he published in St. Vladimir's Theological Quarterly (Vol. 28, No. 4), "The Paschalion: An Icon of Time".

The situation was further complicated by the introduction of a new calendar in the Christian west by Pope Gregory XIII in 1582, which resulted in a completely different calculation of the date of Easter by the Roman Catholic Church.  The Orthodox east declined to adopt the new calendar for a number of reasons, one of which was the fact that it violated Apostolic Canon VII, since it allowed Easter to occasionally coincide with the Jewish Passover.  

Due to the introduction of the Gregorian calendar, Orthodox and Roman Catholic Easter now only coincide once every several years.  Most Orthodox jurisdictions that have adopted the Gregorian calendar for practical, civil reasons still celebrate Great Lent, Pascha (Easter), and Pentecost on the same dates as those on the Julian calendar, in order to comply with the decision of the 1st Nicene Council.