What order are you turning on your devices? Try turning on the AVR and TV first, then turning on the computer.
And I dont think splitting the HDMI's up will work, you will get video but you will unlikely get audio since hdmi audio needs a functioning video signal to work. The cable going to the AVR wont pass HDCP and any protected content will be blocked.
I have a similar issue with a new Sony receiver that is screwing up the HDMI connection when I wake the computer, the audio stops working or the screen is black. I dont know if its the receiver or the video card (460gtx). If I turn off the receiver while the computer is still on then it usually breaks.