Recommended proximity of HVAC vent to return air

It depends on a few factors but just based off of what you've mentioned I'd say that it probably would be fairly good at keeping the room a more moderate temperature.

This is mostly because the air handling unit will be drawing the hot air out of your room and in the process the cooler air from the rest of the house will displace the hot air. Since you mentioned that the room cools very quickly I'd venture a guess and say that the room has good air flow in and out (air will flow down the path of least resistance) or is very close to the air handler (making the installation of a return simple).

It depends a lot on how easy and/or costly it is to install an air return in the room but it seems like a fairly good idea. The new California residential building codes are actually mandating a whole house vent fan to effectively accomplish the same thing (cooling without needing to run the AC) because it's really effective at reducing the temperature more efficiently.

If you do move forward with installing air return remember that the higher the intake is the hotter the air it draws in will be.

If you shift more airflow upstairs, it's likely it will become noisy. Do not restrict anything unless detailed duct size analysis indicate it is indeed oversized. Insulating ducts will reduce resonant noise, and slightly reduce carried noise such as combustion noise, but it will not stop the whistling and air flow noise at the grille. If you do get insulated ducts, they need to be upsized to make up for the insulation thickness.

To reduce whistling and air flow noise right at the grille, you need a less restrictive grille. Sometimes just a different grille of a different configuration will stop whistling, but not air flow noise. If a less restrictive grille is not possible, the only other solution is to increase the size or add another inlet so there is less demand on the one by the bedrooms.