You've probably done most of this, (several times

) but just to cover all the bases.....
When you plug the USB to serial cable in to your computer, do you get a COM port appearing in device manager? What does it say? Mine says: Prolific USB-to-Serial Comm Port (COMM10).
You mention the software and computer are set to 9600.. But make sure the -radio- is also set to that baud rate. Menu item 56 in the radio should also be set to 9600 bps. (It should be since it's the default).
In MCP-2000, make sure under Radio, you have the correct COM port and Baud rate selected..
Once set up, when you click 'Read' in the program, the PC icon should come on, then the radio should beep and the download should begin. It takes about 25 seconds and you'll get a progress bar on the screen showing the status.
The fact that you see the PC icon means that there is some connectivity between the computer and the radio. I can't think why the icon would just go off without the download occurring.... (FWIW, I just did a full read with both receivers set to memory mode.. But to be safe put them in VFO mode as has been suggested)..
If it's just not working, you might try a partial or full reset. (P 92 of the owner's manual). A full reset will be a pain since it will delete all the settings and memories you've made. And it will be a double pain if everything gets wiped and it doesn't solve the issue
Also.. There have been problems reported with some USB to Serial cables. The ones with the Prolific chips seem to be the best. I can't really add much more on this front..