not sure about the hum, filter cap is the best place to start.
in the 2000GTL, this cap is mounted to the bottom of the chassis, away from the rest of the power supply.
it is a 4700uF 35 volt cap.
you should replace it with a 5600-6800uF cap 35-50 volts.
as for the speaker switch; i will tell you how it is supposed to be, and you can go from there.
turn your radio upside down, with the front of the radio facing you.
take the bottom cover off.
remove any wires that are soldered to the speaker switch that are going to the PLL section. (only remove the wires that were added by the golden screwdriver! not any of the stock ones that are in a nice neat row)
remove the two screws holding the switch assembly to the front of the radio.
now, holding this board up, with the green side (solder side) facing you, the three wires that should be connected to this switch are, from left to right:
yellow, blue, purple.
these are usually soldered to the board from the bottom side, but there is no reason you cant just solder them right to the solder side of the board.
you can see which three pads to solder the wires to by following their traces to the switch. very obvious.
so look around the wire bundle near the switch board for some wires of these colors that are loose, or taped back, and two of them might be soldered together.
they might be tucked away, so look close.
you might also want to check the speaker jacks in the back of the radio.
if you look closely at the jacks on the outside of the radio, you will see small plastic washers isolating the jack's ground from the chassis of the radio.
a check with an ohm meter should confirm that there is no continuity from the jack's body to the metal chassis of the radio.
this should be the case for all three jacks on the back, the headphone jack, and the AUX jack.
good luck,
LC