As far as firewall based stuff goes, I've heard really good things about m0n0wall:
http://www.m0n0.ch/wall/ .. basically a livecd that configures your system as a firewall.. for this application, a livecd is probably a bit better then an actual distro installed to a HD. This way, if by some chance you think the firewall got comprimised, all you need to do is reboot.
Managing multiple computers from one box is probably possible. I only run linux on a few computers, all of which have unique configuarations, so I can't answer that. However, rather the VNC, all you need to do is probably done at the shell, so SSH would be easier, because it would seem that you are actually sitting at the computer there.. its text only though.
OpenOffice is a good choice for office stuff.. past that, anything that does not have a linux port is probably supported under WINE (actually its called some really weird name now, but I don't know what it is). Games will work, yes.. but the best preformance would still be under the native OS (Windows).
Running a firewall + server together may be a good choice, but combineing all 3 may not be the best idea. On the plus side, m0n0wall is targeted at embedded applcations, so it doesn't require anything close to a good computer.