Its not copper ADSL as the copper only carries the phone connection. You have FTTC (VDSL), and there are plenty of VDSL modem/routers that will work on FTTC, it all depends on what your connection speed is.
If you are using a BT HomeHub, you have to login to it to change the SSID. I also recommend that you disable Smart Setup in the Router, as that can be used to identify that you're using a BT HomeHub.
Just tell anybody who asks that the charge is £50 per minute with a minimum fee of five minutes paid in advance although I would personally use a stronger message, the last word being "off".
