could they be arranged in a 'sheet', perhaps within the holes for the shafts,
yes do that :-)

Just make sure there is a small gap around each buttong so they don't stick together.

Most print bureaus that use sls printers will use a minimum of 0.1mm gap between parts.
That means that YOU leave a 0.2mm gap as tha gap is on all sides so 0.1mm per side.

Not sure what cad you use but that button array would be really simple to do in openscad.

About a 4 line script :-)