What is the name of the part on the inside of the door panel that connects the interior pull handle to the cable that releases the door latch? Has anyone replaced this item themselves and how difficult was it to do so?

 

If it is an 01 or newer car you cannot replace that part only. You have to replace the entire door panel.
But usually you can drill small holes by where the cable attaches and route a wire tie to hold the cable in.