That's the MOST RIDICULOUS thing i have EVER heard. I would tell your "Telco" that their device is not the only one. Yes, you maybe have a different one, but they cannot deny you ALL tech support because you changed your stupid router. And I would let them know that if they try to pull one over you.
Yes, you can hook an airport up to a modem. The modem does the DSL/Cable thing for you, the router is a firewall/IP Router that takes the connection and spreads it out.
The first thing I did with my Cox Communications connect was ditch their crappy router and use my airport. The airport works fine, and I was restarting Cox's router like every 5 minutes. If your DSL Modem has a router built in, then you may or may not be stuck. If it is, you can just turn off the firewall on it and take one of the ports and connect it to your Airport, or other router.
Also, you can just try to issue manual IP addresses. In the Router field, enter the IP Address of your router (10.0.1.x or 192.168.0.x) and the subnet as 255.255.255.0