Does every computer that needs to print have to install the Direct Print Client, or only the ones with a printer attached?
Neither — one machine is enough, as long as it can see all the printers you need. We recommend a dedicated PC for the job; a Raspberry Pi is plenty, and that is what we run in our own office.
It doesn't matter whether the printers are on a local or an external network. If the Direct Print Client can see them, you can print to them.