Yes, the receiver has to be on to get picture. You could use HDMI to the TV and optical/coax to the cable box, but if you do this you won't be able to use any of the lossless audio formats (PCM, DD TrueHD, DTS HD) although, none of your equipment listed will do this anyway, unless you have a Blu-ray player you didn't list.
If I'm not mistaken, the HK 247 uses HDMI 1.1. Despite what places like Best Buy will tell you, there is no HDMI 1.1, or 1.3 cable . . . the equipment is what is 1.1 or 1.3. Just get a good HDMI cable (Monoprice.com, Bluejeanscable.com) and it'll do anything you throw at it.
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