"Bridegroom Service" is a popular name given to what is actually the Matins of Holy Monday, Tuesday, and Wednesday. As such it is not a special, distinct service. It is "nothing" more than Matins. (Just like the Matins of Holy Friday - served in anticipation on Holy Thursday evening - is sometimes called the Passion Service or the Reading of the Twelve Gospels.) The name derives from the Tropar (Behold, the Bridegroom comes at midnight) which is sung after the Alleluia. The Alleluia and its accompanying verses replace the usual "God is the Lord" of Matins. The replacement of God is the Lord by the Alleluia occurs on weekdays not only during the Great Fast but also during other Lenten periods (depending on the commemoration of the day).
The Tropar (Behold, the Bridegroom comes at midnight) is based on Matthew 25:1-13.
I have seen, in some churches, a solemn procession during the singing of the Alleluia and Tropar with the Icon of The Bridegroom (also know as the Extreme Humility). However, there is no mention of this custom in the Slavonic Triodion. Also, the Slavonic Triodion gives no special name to this service except "The Matins of Holy Monday (or Tuesday, or Wednesday). I cannot vouch for what the Greek Triodion says.
As for the date and origins of the hymns, that would require someone with a working knowledge of liturgical Greek and access to ancient manuscripts of the service spanning the centuries to compare the changes that occured. However, like most of the service of Matins, the hymns were probably composed at different times and in different places and added to the service over centuries to give us what we know today as the "Bridegroom Service".
I hope this helps.