Is the register where the first marriage was registered still available? I read that the Church had little to do with marriage till the 13th century, when it took legal control of marriage. Then, Church registration followed. 

My questions are: 


Where was the first registration done (i.e., in what country) and what were the names of the persons involved? 
Also, is the register where the first marriage was registered still available?
The source does not explicitly say the first record of the Christian marriage in Europe, but it does imply that the first Christian marriage occurred during the Middle Ages in France, which fits the asker's timeframe.

Source: McDougall, S. (2013). The Making of Marriage in Medieval France. Journal Of Family History, 38(2), 103-121.


  Twelfth-century Northern France offers important lessons for the
  history of marriage. It is a time and place that has been described as
  when and where modern (western) marriage was made. 
  
  Christian, monogamous, indissoluble marriage as the unique source of
  legitimate offspring was a collaborative creation of the men and women
  at the top of the social hierarchy.
  
  Prohibitions were a tool to use against the weak, or to bring down the powerful. 
  Acceptance of Christian marriage law promised salvation, promised sex without sin, and so much more.


Before that time period, the powerful European families allowed polygamous affairs and divorces. The change sparked the formation of the monogamous, indissoluble Christian marriage. The Christian marriage was well-received, because, as the author states, it served useful for their own interests.

The article lists a couple of early recorded Christian marriages, mostly from upper-class people. 


Marie of Boulogne, Abbess of Romsey, married Matthew, younger son of Theoderic, Count of
Flanders.
Prince Henry of England married Princess Margaret of France. 


On this website, it says: 


  First, parish registers - the most familiar modern source for
  genealogists - should not be forgotten, as in principle they can
  extend as far back as 1538. In practice, few survive this early, and
  the information in early registers can be very brief - for example,
  baptism entries may not name the parents, and marriage entries have
  even been known to leave the wife anonymous.


As you can see, the first Christian marriage during the Middle Ages, your timeframe, may be lost forever. Also, few records survive as far back as 1538, a few centuries past your timeframe. Marriage entries have been known to leave the wife anonymous; compare this to the genealogies in the Bible, where many wives are left out.