Is there any way other than alt+tab to start playing music? To clarify I can only control music that is already playing from the Steam overlay(not in Big Picture mode). This seems like kind of basic functionality.
Yes you can,in the very bottom of the steam overlay next to the steam logo you should have links for opening web browser,music and settings.
Clicking on Music link opens up the mini-player from where you can select and play songs from your queue.
As of 22/08/14 if you are opted in steam beta the mini player will have additional browse button which will open up your whole library(queue/playlist) and let you browse it inside the steam overlay.
The bottom links are easy to ignore since they are plain text links,small and near the edge in comparison to other flashy links on the overlay.I have been using steam for years and never did I click on any of those links up until now.