How to Setup Server 2003 to Share Internet Connection

Now the simplest, and most rewarding, thing to do would be to build your self a small domain using your 2k3 as DHCP and DNS servers and go from there.
Use your router as the main network backbone, turn off the DHCP functionality and just use it for the net connection.
With the 2k3 doing the DHCP/DNS you will have visibility of the other computers. You still wouldn't be able to see exactly what they are doing online, as this would require third party monitoring software. As they are using the 2k3 for DHCP/DNS this would be very easy to set up.