dpkg is a low-level command that can only complete the install if the dependencies are already installed. apt-get is a high-level command but doesn't know what to do with a package that it didn't download itself. gdebi has neither limitation.
It seems the only free title (at least on the list I got) is the World of Goo demo. It's available via `steam steam://install/22010` and it works without issues (testing under Arch).
Woah, you weren't kidding about switching away from XMonad first. That is the most broken focus switching I've ever seen -- and that's after I figured out why I couldn't click anything in the window.
Ah, it seems like the pointer coordinates are offset from what Steam expects. If I move windows close to the center of the screen, or if I can find resize the window from an offset edge, it works again. With gnome-shell, fwiw.
Next right click the Steam icon in the Unity launcher and pick Store. Pin the icon to directly pick Store next time.