1. (a) That's right. Unless the agency is irrevocable (i.e. where it is given in the agent's interest), the authority given to an agent by a principal while sane is terminated if the principal is subsequently afflicted by mental incapacity. For the purposes of agency law, a person is considered 'insane' if they are unable to appreciate the nature and quality of their actions.
Because it can be inconvenient in some circumstances for an agency to be terminated, legislation provides for what is called an 'enduring power of attorney'. Such a power referred to as 'enduring' because it remains valid despite the subsequent mental impairment of the principal.
In the present case, there is no suggestion that B has been given an enduring power of attorney, or that the agency is irrevocable, so if A were to become mentally incapacitated, B's agency would come to an end.