Is it consistent to add to ZFC-Reg. the existence of a nonempty set $\mathcal H_\mathcal H$ that is hereditarily equinumerous to itself?
If that is consistent, then is it consistent that we can have $\mathcal H_\mathcal H$ being of any nonempty cardinality? Formally that is:
$\forall x (x \neq \emptyset \implies \exists y \forall z [z \in TC(\{y\}) \Rightarrow z \sim x])$
Where $``TC(x)"$ stands for "transitive closure of $x$" defined in the usual manner, and $`` z \sim x"$ stands for existence of a bijection between $z$ and $x$.
Yes, and the proof is nearly identical to the argument in this answer: just start with an $M_0$ that consists of a set which is hereditarily of some size, and then formally construct a cumulative hierarchy on it. Or, if you want to simultaneously get such hereditarily equinumerous sets of all cardinalities, take $M_0$ to be a class-sized structure which consists of such sets for all cardinalities.
More generally, that construction shows that given any structure $M_0$ in the language of set theory which satisfies extensionality, there is a class model of ZFC without regularity which contains a transitive set that is isomorphic to $M_0$. This works even if $M_0$ is a proper class (take a direct limit of the construction over all set-sized substructures of $M_0$), as long as each element of $M_0$ has only a set of elements (this assumption is needed for Power Set to hold).