Are you sure it is the 3GS ?
picture looks more like the 3G version.
maybe you can post the number off the circuitboard ?
Because there are different version of each model.
picture you posted is from a 3GS, but different from the Topic Starter.
This looks like the board of the popic starter:
In this case it are 2 coils.
Try to put the coils back on.
And in case they are gone, make a jumper on the place of the coil.
thought these are the data lines of the USB.
That should fix the recognition in iTunes.
Only the overheating ......
Is there more damage to the circuitboard?
Maybe you have some more informatiopn on how this problem occured?
was the phone dropped?
did you use a non apple (car)charger.
do you have a multimeter available?
Maybe you can measure the voltage on the spots where the components are missing?
And then repeat the measurement with the usb cable connected to the charger/pc.