But keep in mind that the sound of the cooling will be irrelevant once that user starts typing, as their mechanical keyboard will produce enough sound to not only drown out the fans, but also to shatter their eardrums

So in the end it might be better idea to go for “push” instead of “push/pull” solution?

Putting a fan after another will up pressure (doubling in theory), but not displace more air. Except if the air has to go through something slowing it down where more pressure will help to push more air through it.

There are obviously lots of variables at play (the extra fan takes up space too, reducing airflow for example) so you are surely right that it's probably useless or worse to have a "double fan" for a PC cooling system, but the fans being in perfect sync is just a noise problem (on small fans like this).

This is incorrect. Two fans in series doubles the amount of pressure generated by the fan, while leaving the max flow rate the same. Two fans in parallel doubles the flow rate but leaves the pressure unchanged. This is why some radiators have two fans as it increases the airflow when needing to overcome something with high static pressure.

Fans in series should not be so close as it would cause turbulence, however, some gap will prevent this (or something that naturally fixes turbulent flow, like a radiator).