The app isn't strictly in the background. It is on Windows servers. I am not sure how this is handled on WP7.x, but I do know the apps are different than in WP8. Hope that makes sense.
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It really is very good. And to be available on all three major smartphone OS is awesome. I believe it is unavailable, as an app, on OSX currently, though you can still access things by logging in online. Being that it is available on iOS, I am sure it is coming to OSX.
No it is. I downloaded it to my GS3 from the Play Store but i dont have a Microsoft account so i didnt keep it. Dont need it right now so ill do it again later if i do need it.