I can help with some of it. It appears to me you have no sound because you do not have sound coming from the DVD to the TV. You do appear to have the best connection for video using component. (You can check on the DVD and make sure you are going Component out from the DVD to Component in on the TV) (As I assume there is no HDMI out on the DVD) So now you need to find the audio out in the DVD and connect to audio in to the TV.
It also appears you have one side correct from the DVR, that would be composite out. But I don't understand how or why it goes into UHF/VHF? That cable should be the same on each side - yellow white red and go from Composite out on the DVR to the Composite in on the DVD.
If you have an antenna along with the Dish account, some of the coax makes sense. It should go to the "In" on the DVD, not out.
That should give you these options on the TV;
Ant
HDMI 2 for Satellite
Component for DVD
While watching the DVD on the TV, you should be able to pick these options on the DVD player
Ant for recording off the antenna
Composite for recording from the Satellite receiver.