The simpler question is to study the 2-groupoid $\mathrm{Aut}(\mathsf{Cat})$ of all autoequivalences of the category of (small) categories (i.e. the groupoid of equivalences of categories $\mathsf{Cat} \to \mathsf{Cat}$). This is equivalent to $\mathbb{Z}/2$ considered as a locally discrete 2-groupoid: the terminal object $\mathbb{1}$ is fixed up to unique isomorphism, and the arrow category $\mathbb{2}$ is the unique minimal generator (i.e. it is a generator and no proper subobject is a generator) so also fixed up to isomorphism. Since every category is functorially a colimit of copies of $\mathbb{2}$, once we decide whether our autoequivalence fixes or swaps the two maps $\mathbb{1} \to \mathbb{2}$ we've determined the autoequivalence up to a canonical isomorphism. And of course the nontrivial autoequivalence is the "opposite category" functor $\mathcal{C} \mapsto \mathcal{C}^\mathrm{op}$.
This is a somewhat artificial question, since an autoequivalence of the category $\mathsf{Cat}$ must respect the notion of isomorphism of categories, whereas all we should really want to respect is equivalence of categories. One way to fix this would be to consider $\mathsf{Cat}$ as a 2-category, and ask about the 3-groupoid of autobiequivalences of $\mathsf{Cat}$. But this starts to involve heavy machinery, and the problem will only get worse if we want to go up the categorical ladder. Hence a more "homotopical" approach:
Question: What is the biequivalence type of the 2-groupoid $\mathrm{Aut}(\mathrm{ho}\mathsf{Cat})$ of autoequivalences of $\mathrm{ho}\mathsf{Cat}$, the category of small categories localized at the equivalences of categories? Equivalently, $\mathrm{ho}\mathsf{Cat}$ is the category of small categories modulo the congruence identifying naturally isomorphic functors. In particular, is $\mathcal{C} \mapsto \mathcal{C}^\mathrm{op}$ isomorphic to the identity (presumably the answer is no)? Are there any other autoequivalences, essentially (probably not, but you never know)?
A variant would be to ask about the 2-groupoid of functors $\mathsf{Cat} \to \mathsf{Cat}$ which preserve equivalences of categories and induce equivalences on $\mathrm{ho}\mathsf{Cat}$.
I'm aware that Toën showed in some sense that the $(\infty,1)$-category of $(\infty,1)$ categories has an automorphism group of $\mathbb{Z}/2$, and Barwick and Schommer-Pries generalized this to $(\infty,n)$-categories for larger $n$. This question is meant to be an easier / more elementary warmup to those questions.
The approach that worked for $\mathsf{Cat}$ certainly isn't going to work for $\mathrm{Ho}\mathsf{Cat}$: Freyd showed ("On the concreteness of certain categories." Symposia Mathematica. Vol. 4. 1968-9, pp. 431-56) that $\mathrm{Ho}\mathsf{Cat}$ is not concrete, so it certainly can't have a generating object, never mind a colimit-dense object.
By playing around, I've convinced myself at least that the indiscrete categories (the equivalence class of the terminal category) can be characterized in $\mathrm{Ho}\mathsf{Cat}$ by the fact that all functors into it are isomorphic (consider functors from the walking equalizer diagram to see that such a category is a preorder and functors from the terminal category to see that all objects are isomorphic). So $\mathrm{Ho}\mathsf{Cat}(\mathbb{1},-)$ tells us how many isomorphism classes of objects a category has, and what isomorphism classes of functors are doing to them. But that's about all I've got.