a) Consider the canonical map $B_1\otimes_A B_2\to C$. It is finite ($C$ is finite over $A$) and is surjective on the generic fiber. The spectrum of the lefthand side is a disjoint union of normal schemes because of the étale hypothesis. This implies that the spectrum of $C$ is actually one of these connected components. So $\mathrm{Spec} C\to \mathrm{Spec}(B_1)\times_A \mathrm{Spec}(B_2)$ is a closed immersion. This implies easily a).

b) Counterexample: take two quadratic extensions of $\mathbb Z$ defined by $\sqrt{a}$ and $\sqrt{a+p}$ with $a$ an integer prime to $p$. They have the same residue field at $p$.

[Add] In a), Galois hypothesis is not needed.

a) Consider the canonical map $B_1\otimes_A B_2\to C$. It is finite ($C$ is finite over $A$) and is surjective on the generic fiber. The spectrum of the lefthand side is a disjoint union of normal schemes because of the étale hypothesis. This implies that the spectrum of $C$ is actually one of these connected components. So $\mathrm{Spec} C\to \mathrm{Spec}(B_1)\times_A \mathrm{Spec}(B_2)$ is a closed immersion. This implies easily a).

b) Counterexample: take two quadratic extensions of $\mathbb Z$ defined by $\sqrt{a}$ and $\sqrt{a+p}$ with $a$ an integer prime to $p$. They have the same residue field at $p$.

[Add] In a), Galois hypothesis is not needed. The extensions $\mathbb Z[\sqrt{p}]$ and $\mathbb Z[\sqrt{ap}]$ with $a$ not a square mod $p$, give a non-étale counterexample to a).