The addresses are assigned by your ISP, and are handed out dynamically just like your IPv4 address. If you used the router supplied by BT then it would almost certainly have been preconfigured for IPv6.
Sure, there's almost nothing on IPv6 that isn't also accessible by IPv4, and will be the case for many many years. Web sites are not going to cut off their users.
For the large mobile operators, IPv6 has the advantage of reducing the load on their NAT boxes, and hence improving performance. Although the number of websites accessible by IPv6 is still small, the big ones like Google/Youtube/Facebook/Twitter are on there, and account for a large proportion of traffic volume between them.
For home networks, it can be useful to have IPv6 for accessing remote IPv6 networks, or allowing access to your own servers, without having to mess with VPNs or port forwarding. Some peer-to-peer games work better over IPv6.