Let $b\equiv c\equiv 0$ modulo a prime number $p$. Let $d \equiv 1$ modulo $p$. Say $b, c, d \in \mathbb{Z}/p^e \mathbb{Z}$
Is it clear that there is a unique $a$ in $\mathbb{Z}/p^e \mathbb{Z}$ such that $$ad-bc = 1 \quad ?$$
I do not quite see how uniqueness can be independent of the $p^e$ power chosen.
Yes. Because $\gcd(d, p^e)=1$ so $d$ is invertible and you can solve
$$a = (1+bc)d^{-1}.$$