Same thing happens to me with Ubuntu Raring Ringtail alpha running Gnome Shell 3.7.90 and the nvidia driver. My main monitor is 1920x1200 and my TV is 1920x1080.
It's not impossible to hit buttons and other things in BigScreen, because they highlight when it thinks your mouse is over them. But obviously it's not usable.
I have described my problem here http://steamcommunity.com/app/221410/discussions/0/846940249183612168/
Same issue. Using nVidia. But I can quit Big Picture mode using gamepad.
Also similar bug #921
Fixed. Haven't checked in a long time, but now it's working like expected.
Nothing extracted yet.
When running Steam in Big Picture mode, it always uses the resolution of the primary screen as its "Window Size" setting, even if it is started on another screen.
If the resolution of the primary display is smaller than the one where steam is running, this results in steam being displayed only in part of the screen and it makes mouse input impossible, because the buttons are not where they are displayed.
(screenshot)
I'm using i3 (tiling window manager) on arch 64 bit and nvidia 310.19.
System Information: