I am trying to wrap my head around rings (defined to have unity). I get that in a nontrivial ring, $0$ by definition does not have a multiplicative inverse (so is not a unit). But in the trivial ring $\{0\}$ the multiplicative identity is $0$, so in that case I would say that $0$ does have a multiplicative inverse. And so in this case it actually does make sense to write something like $0 = 0^{-1}= -0$.
Is that correct?
Yes, since $0 = 1$ in the trivial ring we have $0.0 = 0 = 1$ hence $0$ is a unit and is its own inverse.
Also notice that every ring where $0 = 1$ must be trivial since for any element $r$ in such a ring $r = r.1 = r.0 = 0$
And as suggested by a user in the comments, if $0$ has an inverse in a ring, then this ring is trivial too. Call the inverse $x$. Then $0 = 0.x = 1$ hence by what I wrote earlier the ring is trivial.