The AEBS likes DHCP, static is a cost that you do not need. If the modem is faulty then they should replace it but the AE will support 10 computers, Windows and Macs, wirelessly.
The AE should have an ethernet port that you connect to the modem, do that and reset the modem to DHCP, power it down and then back up, this will handshake with the AE and assign an IP address which the AE use and give each computer a DHCP address.
My setup is a cable modem, Airport Extreme and an 8 port ethernet hub, the modem supplies the IP from the outside world, the Airport splits it to the switch and other wired devices, it also assigns IPs to all my wireless machines.
I did it just as I described above.
The ethernet port on the modem should be the out to computer, the others are used by the built in router, you don't need them.
Then to the AE, it will do the rest, you will have to set up SSID etc on the AE if it is not already.