Re: What I would like to know.
Regarding music purchases from Google Play Music...
When you buy a song (or album) from GPM it is added into your Google Play Music account. If you open the Google Play Music website on your computer you'll find all of the songs in your library are there, and next to each is a little menu button (three dots). Click this and there is a "download" option. Alternatively if you have the Google Play Music Manager installed on your computer for uploading your music collection to your Google library there is a "download purchases" button in there.
How do I know this? Because I use it all the time, and if I buy a song from Google I sometimes want to play it in my car on my iPod, so I download the MP3, add it to iTunes, and transfer it through there.
FWIW you can also download iTunes purchases and play them on anything too, as they're just MP3s after all.