If Yahoo need any extra verification, it will be requested the next time you log in.
They - and the other e-mail majors will NEVER send "click to confirm" links unsolicited
As to the BY / Yahoo thing - BT are simply too small to survive on their own. They needed the resources of Yahoo to provide content, advertising and hosting
However its interesting that BT are currently in the process of moving their business customers to Microsoft's servers - and at the same time taking the advantage of Microsofts "cloud" system to offer hosted Office 365 and hosted exchange mail. If you trust them with your business data its actually quite a good deal.
As I said...IF you trust them with your data