1, What Linux version for the Server?
Debian. Free, very clean and easy to maintain/upgrade/cleanup.
2, Windows or Linux for the Workstations?
Debian (or Knoppix) and OpenOffice.org (a free fork of StarOffice) which is also free. Knoppix is easier to install and geared towards desktop systems but is based on Debian so the packages they use are mostly compatible with eachother or in most cases identical.
3, Compatibility issues for DOS Software?.
dosemu, dosbox, wine, win4lin (which runs win95/98/me on top of Linux), vmware... at least one of them will do. The first three are free and the last two are commercial and cost money.
4, How tight could the budget be on a project like this?
You have the hardware, all you need to do is download a Debian ISO CD-image, burn it on a CD and start installing.
BTW: If you want a more responsive system and get a system that is as fast as possible make sure you install Debian or Knoppix with the Linux 2.6 kernel (linux26 or knoppix26 as boot options).
Sarge netinst Debian Installer beta4 (I recommend you try this first, more up to date)
Offical Debian CD-images