As far as the original question ("Can you run an AppleTV without a display?") I'd double check first. I just tried turning the TV off, while leaving the A/V receiver and AppleTV on - and got nothing. It may be that the AppleTV needs a signal from the HDMI connection that it is attached to a powered-on TV.
I'd experiment further, but having recently spent many, many hours getting all my connections just so, I'm hesitant to start playing around with cables.