What are each of the knobs for? presumably volume, balance, and ?
Treble, Bass, and Volume perhaps? Just guessing here myself.
De-soldering large components isn't hard, especially the pots. I constantly remove and replace 15-pin VGA connectors at work from circuit boards with no ill effect; certainly a 3- or 4-pin pot is a no-brainer. A solder sucker will take care of those no problem. I do agree that practice should be done on a junk board first if one has never done anything like this but really, how hard is it to melt a solder point and press the button on the sucker?
As for the rectangular hole, I can't make too many suggestions because I don't know what went in there originally and what they want to put in there now.
Personally, here's what I'd do if I wanted to avoid electrical work:
-Put the circuit board as-is, behind the rectangular hole.
-Fill rectangular hole with a panel, perhaps covered with speaker fabric for aesthetic reasons. Hinge it so you can get to the original controls. If originality is no concern, then put holes in the panel to line up with the existing board's controls.
-Install "dummy" knobs on the two original holes with glue, epoxy, etc. or attach only one knob for volume (see below)
-Alternatively, the board can be mounted so that the controls and plugs are behind the radio.
-Speakers can go where you want, where they fit.
Lots of options and if you're crafty with tools soldering might not even be needed. At most you'd need to extend the speaker wires and you can use crimps for those.
If your son really insists on moving the knob(s), then the potentiometers (pots) will have to be de-soldered and extended using wires and more soldering. I can't see too well in the pics, but they look threaded and you'll need the nuts for them. The holes on the old radio would then need a smaller hole for the pots. You could probably find a washer that has the right ID to fit the pots through, yet a large enough OD to cover the large hole from behind.