IMHO, in your example "des" is not used as a short form "deshalb" or "deswegen", but as a synonym to "dessen".
Here is a bit more of the actual songtext:
Als ich bei meinen Schafen wacht
Ein Engel mir die Botschaft bracht.
Des bin ich froh, bin ich froh,
Froh, froh, froh, o, o, o!
Duden uses another example:
Des (dessen) bin ich sicher
So I would say in this case it's used as the genitive singular, standing for "die vom Engel überbrachte Botschaft".
Also you should keep in mind, that this is quite an old song (written around 1600) and that language is constantly changing.
As far as I know, this figure of speech isn't commonly used anymore.
To answer the question if "darum", "deswegen" and "deshalb" are interchangeable, the Duden lists them as synonyms.
I wouldn't know about a case where these three words could not be replaced by one another.