It is recommended to post logic questions in the Discrete Math subforum.

[(A -> B) /\ (~A -> B)] <-> B is a tautology, so we can conclude B from A and conclude B from ~A iff we actually don't need any assumptions and can conclude B outright.

You can conclude B from A because A together with ~A, which is true, gives a contradiction, which implies everything. You can't conclude B from ~A because, since ~A is true, B would also be true and S would not be valid (since it contains ~B).

I am not sure I understand the question. What does "true in A" mean?