I don't know the current state and never used it much, but you can specify local folders with music that it will play, but it doesn't upload tracks to the cloud, and I'm not sure how well it works with search.
Yes, if you do this on the desktop app they'll transfer to your phone's local storage over wifi. It's so much easier than trying to figure out how to get music onto an iPhone the Apple way.
I think this is what is meant by a “music library”