OK, let me start my answer to your question by saying that nothing is wrong with your set or cable box. As a matter of fact everything is working really well and that is why your picture quality can be great or not so great. High definition will do a state of the art reproduction of any broadcast/cable /satellite content it delivers to your screen in the form of a picture. IT will do a great job with a new movie or a silent movie from 75 years ago , HD will show how great anything that's been produced in HD will look, and conversely will show just how poor the quality of a 75 year old Charlie Chaplin silent film will look ,what you are seeing is how both the old movie and newest movie REALLY LOOK FOR THE FIRST TIME. So the HD is really something to see , the details and the rich colors ,surround sound etc. what about the old silent film, HD will show in great detail just how defective the film print is ,scratched nicks ,faded print etc. .To boil it down to gravy , WHAT YOU SEE HD IS WHAT YOU GET . That is one answer to three parts of your question
Next, yes you have an HDTV cable box and it is digital, DIGITAL IS NOT HDTV . It is, a way of sending the signal to your box by converting the information that makes up the picture (and surround sound) into binary code zeros and ones , instead of an analog signal , what that does is permit for a faithful reproduction of what is sent over the wire cable or air satellite. What gets sent is what is delivered without any lost information -and that is the purpose of digital.
Finally, what can be done to improve the picture, for the older stuff that is not HD , not a whole lot. if t looks bad HD will just show how bad it is , after 02/19/09 ALL TV will be broadcast in digital and that will improve the signal, but not poor quality. Comedy channel etc. will go to HD over time, so improvement will come eventually . Now this may or may not work depending on where you live. If you live in or near a large city , high definition broadcast over the air television channels ABC CBS NBC FOX ,AND LOCAL STATIONS will be available to you by the use of a good old fashioned over the air TV antenna , and here is something the cable and satellite companies will not tell you about over the air HD reception , IT CAN BE BETTER QUALITY THAN CABLE OR SATILLITE . WHY? In order to send so many channels to a cable or satellite box the signal has to be squeezed and compressed and this degrades the picture quality a bit, most folks would not notice or care. This only applies to HD over the air AND NOT NON HD so you could only improve the HD a bit and nothing else. I was a child in the 1950's and all we had were water powered (kidding) TV sets that had small screens and no color and looked like the picture was being watched through a poorly focused fish eye lens at times, so even though HD is not always perfect,I am amazed by it at times and can't be spoiled by it enough, I hope this answers your question even though I can only offer you the limited solutions that are available ,read on there is a bit more .
If you are not using an HDMI CABLE<BEST FOR PICTURE AND SURROUND SOUND IN ONE CABLE , YOU ARE NOT GETTING HD PICTURE AND SURROUND FROM YOUR CABLE BOX TO YOUR TV . COMPONENT cable is good for the picture only (blue,green,red- wire) plus add stereo left and right channel sound (white, red,wire) = five wires NEXT DVI -one cable plus left and right sound channels or DIGITAL OPTICAL (fiber optic) cable for surround sound (to receiver) from cable box . = 2 cables . IF you have any questions about the various hookups post them here or sent a message , I will be happy to help . Good luck