I would run a HDMI connection to the TV, and pull sound through a Sound bar, for when you are not using the AV receiver, and if wanting sound through the AVRS, use it as the switcher. So from the closet, you would have two Coax runs for Sound for the Sound Bar if you decide to go that route, One HDMI, which can either be a pre-terminated HDMI cable, or Cat-6. Switch everything through the AVRS if you go how I did below.
I have my Blu-ray only sending sound through our Denon 1612, since I only wanted to use our Sound Bar, when watching tv programming, that is not Sports or movies. I have basically a stereo cable for sound from our UVerse DVR to the Sound bar, a single HDMI from the AVRS to the TV, DVR & BD through the AVRS for routing.