I purchased a song from the Google music app and it stores to cloud player... It won't give me the option to stir to device unless I activate account... an I mistaken that if I link my info with Google play music it will charge me... So then I am paying for song and for Google music?
You are mistaken. It will not charge you to use the app unless you sign up for the service. I have never paid for google music access and have the play music app on my phone and have downloaded all music to device.
Just adding that once downloaded to device, you will have full access in the play music app. If you wanted to play them in another app or take them off the device, they are not saved by the song name, but by a random filename. But you can download the songs, without paying for service as well, on your computer directly from google music so no need to transfer from your phone anyway.