Ad far as cleaning off the circuit board goes, brush on some mineral spirits on the bottom side only, wait a few minutes, then try and brush it off. Lather, rinse, repeat so to speak. It should remove the old adhesive without harming anything. And if it doesn't, don't worry. I don't think it will hurt the functionality of the board, whether it has this old foam gunk on it or not.
As far as re-soldering the wires go, cut and strip the wires for new bare ends. Pre-tin them with solder. Insert them through the HOLES in the electrical pickups (and NOT next to the rivets), and solder from there. Use a piece of water-soaked sponge to place on the pickups where they touch the plastic, to act as a heat sink. Immediately when done soldering, wipe the wet sponge over the soldered joint in order to kill the heat. This should keep the plastic from melting. I do this all the time, and it works great.