Install a file manager like File Expert and do a search. That will help you track them down.

 

When you purchase music from play store, I believe it doesn't automatically download it to your device. You can go into the play music app, and choose to download it to your phone of course, but remember that the play music downloads to the system partition (if you are rooted, that's no issue)

The easiest option would be to just connect to the play store on your PC and download it (and then transfer over to your phone via any of the sharing options available - dropbox/box/drive or just connect using USB cable or wirelessly)

Yes, Google prefers that users use their app, rather than make the downloaded music available to other apps