For $z \in \mathbb{C}\setminus \{0\}$, is $|1/z| = 1/|z|$?

Yes.

Note that $1/|z|$ is, by definition, the real multiplicative inverse of $|z|$, so by uniqueness of such an inverse, since

$$|1/z|\times |z|=|z/z|=1,$$

we have $|1/z|=1/|z|$.