" Can I just connect another usb broadband modem to it and use it as I do my main PC? "
Yes. But obviously you can only use one modem at a time. So it would be one machine connected to the net and one not connected. If you use a rj45 crossover cable and connected it to your network socket on both machines. You can share the internet connection.
The bog standard telephone line sounds as if it's working fine.