What should I do to make f-pin coax coupler I bought (to link two coax cables to extend the range of my cable box) receive and display cable programs?

Bad cable! How does TV attach to the satellite receiver? Coax, RGB, etc:? Try another cable first and foremost. If the cable is DB15 one bent pin in connector will cause exactly what you have described. Always check cables first!

if you have a coax digital out on your tv, why not use that, you have coax, optical, and hdmi inputs on your reciever, but the easiest, low cost way is the coax

Replacing connectors on televisions can be a pricy endeavor. If you can't afford to bring it into a service shop, you may have luck by switching to a different type of connector. If your Satellite or Cable receiver has an HDMI output, your best bet would be to buy an HDMI cable and hook that up instead - the quality would actually be better than coax. Failing that, depending on what kinds of outputs your sat/cable receiver has, you could also go with (in order of preference) component video (a cable with five ends, red/green/blue, plus two for audio), s-video (a round metal connector with four pins in the middle), or composite video (similar to a component video cable, but only three ends - red, white, and yellow).
Any of the above cables (including the HDMI) can usually be found at your local big box electronics retailer for under $30.

Yes, mostly all new displays are at least 720p which display a great picture ! However, coax cable is the worst possible cable you can use. Your best bet for HD from the cable box is to use component cable. These are the 3 RCA type colored cables, red, green and blue. You will also need another two RCA cables for the audio, which is usually red/black or red/white. If you have a surround sound receiver, then you will not need the 2 for audio to the TV. These 2 would goto the receiver or you can use a better digital coax or optical cable for audio.
Also, HDMI from the cable box to display is a more expensive cable which will give you HD but doesn't always look better then component on some HD cable boxes.

1. Video-out to TV: You should be able to connect the receiver to the TV with any / all of the available video modes.
However, if you don't have the HDMI connection, then the Video-In on the TV would have to be switched back-n-forth between Composite / S-Video, depending on how each device is connected to the receiver. [If the TV has 'auto-sensing' Video-In then it should automatically detect the incoming signal from the receiver whether S-Video or Composite or Component.]

2. Audio-Input to receiver: As the receiver provides only two Coax-Inputs, you would have to first assign Coax-1 to DVD and Coax-2 to Cable. Then connect the coax cables accordingly; that should do the trick and allow you to hear the cable sound over coax.
If you still have a problem then check the 'Input mode' setting; leaving it on Coax-1/2 (Auto) would be the better option. This is on pg. 50 of my manual "Specifying the Digital Signal Format".

Usually if you have a Cable or Satellite box, you would hook up a Fiber Optic or Digital Coax cable from the Sat/Cable box to the Sony Receiver. Then for HD, an HDMI cable or componet cables from the Sat/Cable box to the TV

You can use the yellow RCA cable using an SVideo adaptor. When using dual displays make sure that display number 2 is adtive. To do that move the mouse to display 2 press the right mouse button and select active. Check extend desktop to the monitor to check the signal and press apply. Make sure you are using either an Svideo or RCA signal.

the avr-588 apparently cannot carry sound via hdmi - don't know why; just what i found out from the geek squad at best buy; solved the problem by connecting optical cable from cable box to receiver; be sure to change the input on the receiver for opt-2 (system setup >2 inputs) to correlate with tv