Agreed on all counts. "Hallelujah" has taken on something of the status of a secular hymn, and I'm guessing that's why it was chosen, even if it's mainly about sex. (You can just listen to the "hallelujah" part.) I think John Cale said that when he asked Cohen if he could cover the song, Cohen sent him enough pages of lyrics to fill the Sunday NYT, so there's a lot from which to pick and choose.