Wow, you've got almost the same setup I have right now, only instead of a H/K receiver, I've got a Marantz. I just recently stumbled through the same troubles you're having.
I wouldn't say you're out $25 for a digital optical cable, and the info about using regular RCA cables as 'backup' sounds like nonsense to me. It sounds like it's just a matter of mapping the correct digital sound input to the correct function selection on your receiver. I'm not familiar at all with H/K, but I know mapping the digital inputs on my Marantz was pretty confusing at first.
Ideally, you should have two cables running from your cablebox. HDMI to your TV, optical to your receiver, excluding the power cable and coaxial cable from your wall outlet.
You should check the manual that came with the remote from your cable company (if one was included). I was able to program my cablebox remote to control my TV, cablebox, and my receiver all at the same time. This remote can be set to change volume only on the receiver even while in other modes.
I'm pretty new to this stuff too, so if anyone else has some helpful experience, chime in!