I didn't actually look at the sound system directly - though any audio I tell you to send to the TV can actually go directly to the receiver and you can use that instead.
Now, I just looked your manual for the TV up online:
http://merchant.satisfusion.com/library/manuals/owners/51_57F510.pdf
The back of your TV is shown on page 9 - Let's start there.
Follow my instructions and hook cable up to input 1 on your TV with component cables
Input 2 should come from the DVD-R to the TV
Input 3 should come from the VCR
Now, your cable (COAX) can come from the wall, to the VCR, then loop through the VCR to the TV, then from the TV to the new cable box. Or to the TV first, then the VCR, then the cable box.
That should allow you to watch and switch between all the video sources directly on your TV.
Now, follow my instructions above to allow you to record from the cable box to the DVD-R and from the VCR to the DVD-R.
NOTE: Depending on what technology is in the current Motorola HD-DVR you may have some issues when dumping from the cable box to your DVD-R. I couldn't be sure without testing it myself, but the hookups are still correct.
You may be able to get a haflway decent tech over to your home for a couple of hourse of work at about $50.00 - $75.00 per hour. More or less depending on your location. Where are you located?