Ahh.
Ok, if the wires you are talking about are attatched to metal strips on the base of the power brick then those are the grounds and problems there aren't uncommon. The best way to deal with those are to cut off the pin connectors and group the wires together into small bundles and attatch (solder) lugs (ring terminals) onto them and bolt them directly to the base of the supply
The burned out launcher coil is probably due to a shorted transistor on the driver board. Replacing the board isn't usually needed, they are generally easy to fix. Here's where you will either need a schematic or be able to trace the wires back from the solenoid to the board and determine which transistor. It's should be one of the MPS45's, but since there's about two dozen of them you'll need to determine which one is for the burnt coil in question and replace it before you replace the coil. Which transistor drove what varied somewhat from game to game so you should have some reference to look at when you dig into this. Gottliebs manuals generally had very good schematics and charts for 80x games, do you have a manual?
For the drop targets you could have one of several problems. Does the bank feel like it's trying to reset? I've had several that would bind slightly and prevent it from resetting, that's usually a mechanical problem and easy to fix by taking the solenoid off and checking the slide on the reset arm., it may require slight straightning. If it's not even trying, the drop targets may not be registering closed switches and thus the cpu doesn't know to reset the bank. Reset the bank by hand and either start a game and hit the targets to see if they register/score or go into test mode and see if the switches read properly. Loose/bad solder joints on the drop target switches are common or you could have a problem with a switch matrix diode or one of the logic chips on the cpu board. If the switches do read, double check all coil connections and driver board connections, enter test mode and see if the solenoid fires during the solenoid test sequence. If it doesn't look for an open transistor on the driver board.
The sound is tricky to call from here. The board beeps when it starts up, if it beeps at other times it means it's resetting. This could be a sound board problem or a cpu board problem and would require some hands-on troubleshooting. If you know anybody else with an 80b system, it might be worthwhile to try the soundboard in another machine. If you really get stuck you could send me the board.
Now, having said all of that, 80x systems are notorious for ground problems and poor grounds can make the boards unstable due to differences in ground potential. If you haven't checked out the ground fixes here:
http://www.marvin3m.com/sys80/index.htm then you must. These "fixes" are mandatory for reliable 80x operation and could be contributing to the problems you are having. Since you appear to have ground problems already, you might want to start there and get that right out of the way.
D