1) Will I be able to operate both routers simultaneously from the same BT telephone line?
(My reason for doing this is to be certain that the new router is connected and working correctly before disconnecting the old one.)
Simultaneously as in both plugged in to phone line and powered on at the same time ? No.
Simultaneously as in plugging in one or the other continuously you are happy ? Yes, no problem.
2) How can I discover the ADSL password?
(I have a possible one in mind, but if it is incorrect, will I be logged-out from my current router - and, thereby, disconnected from the Internet?)
Ask your ISP ? And see answer to 1 in terms of answer to disconnecting, as long as you don't tamper with the settings on the "good" box, all it should need is a reboot.
3) What other settings do I need to know when setting-up the new router?
Try the defaults first before fiddling, in most cases the default MUX etc. settings work out of the box. This link may also be of some interest :
Connecting a Belkin Modem/Router to Virgin :: Technical Issues :: think broadband