The first question has an answer. If $p$ is an odd prime, $p \nmid a$ and $p \mid (a^n-1)$, then $p$ is Wieferich to the base $a$ if and only if $\nu_p(a^n-1) \ge \nu_p(n)+2$ which follows from an identity :

$\nu_p(a^n-1)=\nu_p(n)+\nu_p(a^{p-1}-1),$

which is (1.1) in https://arxiv.org/abs/2112.04173 (apologies for self promotion).

The first question has an answer. If $p$ is an odd prime, $p \nmid a$ and $p \mid (a^n-1)$, then $p$ is Wieferich to the base $a$ if and only if $\nu_p(a^n-1) \ge \nu_p(n)+2$ which follows from an identity :

$\nu_p(a^n-1)=\nu_p(n)+\nu_p(a^{p-1}-1),$

which is (1.1) in https://arxiv.org/abs/2112.04173.