Right no " little fall of rain", I'm not going to see it.
A Little Fall of Rain is in it, and it's excellent btw one of my favourite scenes, it's just cut down. You can just about tell where it's cut as well so I think the full song was definitely filmed as well, so hopefully we will get to see it at one point. I imagine the full versions of most if not all the songs were filmed, if we're to believe that the first cut was 4 hours.
As for it being sung-through, you don't need to know much about the musical, or even research it, to know that. How is calling it a musical misselling it? A musical implies that the cast/characters sing, it can be anything from a few songs to singing every line.