Herbigny,
I am no authority so my answers are at best, uneducated guesses based on what I remember, but here goes

These songs are referred to as "Dukhovni Stikhi" (Spiritual Hymns??) and are sung as coloquial folk songs. Of course, they are never sung at church within a service. Perhaps at a lunch, or gathering, hence the inclusion of rhyme and refrains.
This tradition is quite strong. Most Old Ritualists identify themselves as a 'musical' people. The 'Old Believers' (how I refer to bezpopovtsi, to differentiate them from Old Ritualists) are well known for their musical heritage and often peform at folk festivals, etc.
I am sure that this type of song isn't solely 'Old Ritualist' though I haven't heard other communities sing them. I guess as they are quite old fashioned and not very 'theatrical' they fell out of common use. Perhaps some of the Carpatho-Rusyns on the board could advise as to whether this tradiion has survived in their culture?
Cheers,
PS: I guess it is Thanksgiving now? Greetings to all our North American friends for this holiday!
