I think there is no easy way to monitor the current state of a relay, except by connecting another relay to the relay to be monitored. Then you could use the output of the monitoring relays to tell the Arduino their state. I think you don't even need USB or Ethernet since it's an electrical signal you can feed to an analog input (with due caution).
As for alternatives, it all depends on what you can get from the control unit - can you do more than just switch on a relay when an alarm condition occurs? Say, execute a script to compose a message, open a (serial, ethernet) connection, send the message.
I assume that your home, with the Linux server, is also the place where the alarm system is installed. If so, USB should be fine for Arduino-PC communications.