Which properties of categories are preserved under homotopy equivalences?

Question

This question arose from a discussion in the comment section of another MathOverflow question with Mike Shulman and Alec Rhea, which raised the following point:

Vague Question: Are adjunctions an appropriate notion of "sameness" for categories?

While viewing adjunctions as such might seem strange, there seem to be a few reasons to adopt this point of view:

1. They are in a sense a weakening of equivalences: any equivalence of categories $F,G\colon\mathcal{C}\rightleftarrows\mathcal{D}$ can be made into an adjoint equivalence $(F,G,\eta,\epsilon)\colon\mathcal{C}\rightleftarrows\mathcal{D}$, and these are in turn precisely those adjunctions whose co/units are natural isomorphisms.
2. Secondly, having an adjunction between categories $\mathcal{C}$ and $\mathcal{D}$ implies that $\mathrm{N}_{\bullet}(\mathcal{C})$ and $\mathrm{N}_{\bullet}(\mathcal{D})$ are homotopy equivalent as simplicial sets.

So, given a functor $F\colon\mathcal{C}\longrightarrow\mathcal{D}$, we have the following chain of strict implications $(1)\Rightarrow(2)\Rightarrow(3)$:

1. $F$ admits a left or a right adjoint $G\colon\mathcal{D}\longrightarrow\mathcal{C}$.
2. Taking nerves induces a simplicial homotopy equivalence $\mathrm{N}_{\bullet}(F)\colon\mathrm{N}_{\bullet}(\mathcal{C})\longrightarrow\mathrm{N}_{\bullet}(\mathcal{D})$.
3. Taking nerves induces a simplicial weak homotopy equivalence $\mathrm{N}_{\bullet}(F)\colon\mathrm{N}_{\bullet}(\mathcal{C})\longrightarrow\mathrm{N}_{\bullet}(\mathcal{D})$. That is, $F$ is a Thomason equivalence.

In particular, the difference between conditions $(1)$ and $(2)$ is that adjoint functors require not only the existence of natural transformations $\eta\colon\mathrm{id}_{\mathcal{C}}\Rightarrow G\circ F$ and $\epsilon\colon F\circ G\Rightarrow\mathrm{id}_{\mathcal{D}}$, corresponding to simplicial homotopies $\mathrm{N}_{\bullet}(\mathrm{id}_{\mathcal{C}})\Rightarrow\mathrm{N}_{\bullet}(G\circ F)$ and $\mathrm{N}_{\bullet}(F\circ G)\Rightarrow\mathrm{N}_{\bullet}(\mathrm{id}_{\mathcal{D}})$, but also that these be coherent natural transformations/homotopies in the sense that they satisfy the triangle equalities. So we could perhaps summarise these conditions as follows:

1. $F$ is a "coherent homotopy" equivalence.
2. $F$ is a homotopy equivalence.
3. $F$ is a weak homotopy equivalence.

In trying to assess the differences between these, one may consider the following question:

Question. Which properties of categories are preserved under the above conditions?

That is, given categories $\mathcal{C}$ and $\mathcal{D}$ such that we have a (coherent, weak) homotopy equivalence of categories $\mathcal{C}\to\mathcal{D}$, what are some properties of $\mathcal{C}$ that $\mathcal{D}$ ends up possessing as well, or vice versa?