Sometimes, an HDMI connection will provide some rudimentary audio to the TV. If that's your case, then Mr Ben is correct. You won't need to run that RCA cable. Try it with both the cable box and DVD player.
If there's no audio, then you do need to run those RCA cables to the TV.
Again, you'll select your video via the TV and your audio via the receiver.
As for connecting the AVR to the TV, since all video will be fed directly to the TV, not through the AVR, you'll only need that for it's setup. A composite video cable should suffice.