As Buildersmate says, NO Airline including BA has the authority to summarily deny a passenger entry to a country, or through a transfer process. It is the immigration authorities for each country who does this by setting clear rules about what papers you need to enter. The airline is required as a condition of operation, to ensure all passengers have the correct documents as required for their destination, and gets fined if they let people throught incorrectly.
It sounds like the BA staff member saw you did not have the right documents and was therefore unable to let you fly. Just the same as they wouldnt let me fly out of the UK if i forgot my passport.
The passenger rights and Terms of purchase do state it is a passengers responsibility to ensure they have all the correct paperwork and visas for their journey, and the airline cannot be held responsible.
In terms of what to do with the tickets, If you have bought a changeable ticket of some description, you may be able to stil use it subject to a change fee. If it is a non changeable ticket, you cannot, but you could claim the fees and taxes back inus a service charge. - that is the same on most airlines. It doesnt really make any difference if the coupons were torn out or not as long as you still have them. The only way to tell if they can be changed is to take them to a BA ticket desk, or call the BA reservations number for your country who can look up your booking.
Hope that helps.