Connecting all computers to a switch then to a server- can it be done?

As others have mentioned. the server in your design is acting as a gateway(router). and it requires two nic to function (1 to modem, 1 to switch). you will have to configure network services such as DHCP, DNS, firewall, NAT...etc on the server OS.

As for the wireless router, you may have one of its LAN port connected to the switch, and disable all routing functions (NAT, DHCP...etc) then it will basically serve as a wireless AP, (the server still being the gateway/router

it is not recommended to run anything other than network services on the server. therefore i wont recommend running file server on the server. since that any interruption or failure to the server means total network outage.

Alternatively, you may use a ordinary SOHO router to serve as gateway, and connect router with switch. then all PC & the server goes to the switch. that way you are safe to use server as file server