Does every computer in the company that needs to print, need to install the Direct Print Client app on the local computer? Or only the computers where the printer is physically attached?

It’s enough to have only one machine that has access to all the printers needed. We even recommend setting up a separate PC for this. E.g. We have configured a Raspberry PI 4 in our office for printing purposes. It absolutely doesn’t matter where the printers are connected to a local or external network. If the Direct Print Client client sees them, you can print.