I want to say no to your question about playing in Android Music Player once you've downloaded the music, but I am not 100% sure on this. Perhaps someone else can chime in. I was unable to find anything about this in the search.
You can only play the tracks via Spotify. The download function is there to enable offline access to the tracks; mainly so people can download when connected to WiFi rather than using their mobile data allowances, as well as ensuring you do not have to rely on internet connection when you're out and about