No, that connector will not cause the eyes in the skull not to light. That connector is for GI lighting. That being said, that connector and associated pins needs to be repaired. When replacing the pins on the board, pay very careful attention to the traces on both sides of the board. The heat in that area (that caused the connector to burn) makes the traces very delicate, even though they are big...

After you replace the pins, you can use the table in the "On Board GI Problems" section of PinWiki to verify connectivity http://pinwiki.com/wiki/index.php?title=Williams_WPC#General_Illumination_Problems

For the skull lamps, try flipping the LEDs around in the socket (assuming you installed LEDs).

Take a close look at the picture Zaza posted. There are 4 yellow with white stripe and 3 solid yellow wires coming up in that bundle. Each wire of each color is electrically the same (multiple wires coming up from the 6.3v winding on the transformer) so you do not need to find the specific wire that was on each pin. Just match the colors to the correct pins on the connector as shown in Zaza's picture.