If $a(x)$ and $b(x)$ are coprime in $\mathbb{Z}_n[x]$ and $a(x) \mid f(x), b(x) \mid f(x)$ in $\mathbb{Z}_n[x]$, then is it true that $a(x)b(x) \mid f(x)$ in $\mathbb{Z}_n[x]$?
2. You have $ac+bd=1$ for some $c,d$. Write $f=aa'=bb'$, which you may, by assumption. Then we have $f=bb'ac+aa'bd = ab(b'c+a'd)$, so $ab|f$. You can see that this is, in fact, true in any commutative ring.