if at all possible, and should you have experience, i would suggest recreating the anchor traces and using a new part.
Otherwise;
-start by adding your own solder to the pins that are damaged on the board, and as well to the part-pins on the port.
-crazy glue the port onto the board, lined up by the pins, to recreate the anchor trace effect.
-heat up the pins with your iron, and the solder should flow itself onto the other part of the pins.
-add solder to the half of the pins not being heated as required. (Carefully and slowly)
Yes, all pins should be soldered.
this repair is very straight forward, but requires experience to flow the solder correctly. I wish you the best of luck in your attempt.