Music purchased from Google Play is stored on my phone as untagged, numbered MP3 files. Is there a way to get the information Play has about these MP3 files, then rename them and set the ID3 tag so that other applications can easily use these files?
The Official Google Play Music app will download music as untagged, numbered MP3 files, as you said, and uses an internal mapping method to know which MP3 is which.
However, the app Offline Music Importer should do what you want. It is free for the first 50 songs; after 50 it costs $2.49 to unlock.
A description from their app page: