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:
Why you need this app:
When you make your Google Play Music media available offline, you are storing cached music files in a special folder on your device. The media provider service within Android can’t see the music in there. Even if it could, the file names are meaningless and they don’t have any ID3 tags. The service would not know what the song title is, what album it belongs to, what track number it is, not even the artist. The files are just raw music. For all those reasons, you can’t play those files properly in any other music player except the Google Play Music player.
This app gathers those meaningless music files, applies the proper ID3 tags and artwork, moves the files to your /Music folder on the sdcard, and updates the Media Provider. All your imported music will be available to any music player installed on the device.