I assume the ouput says the laptop is disconnected from the internet because the Ethernet cable is removed?
Yes, but that's fine because you are correctly connected by wifi.
In my experience BTinternet and its aliases always supply usernames and passwords for the ADSL connection itself (over and above an email address and password combination for each user). The exception I think is you have BT Connect (a business product).
If you really haven't got ADSL credentials to supply the modem you might be able to set to it be a "modem only". Not sure if your model supports this - the one I saw the other day did.
Though the fact that you were told by BT to use PPOA makes me suspect that you should have a letter from them with some specific ADSL logon details in the format described in my previous post.
You may need to call them to verify.
hth,
amofw