Yes. If you start with infinitely many urelements and then take the sets whose kernel is finite (the kernel of a set is the set of urelements in its transitive closure), the resultant inner model will satisfy Replacement. A more generalized argument is included in my dissertation (https://arxiv.org/abs/2303.14274, Theorem 26).

Let ZFU$_\text{R}$ denote the ZF set theory with urelements axiomatized with Replacement. In general, given a model $U$ of ZFU$_R$ an ideal $\mathcal{I}$ of the class of all urelements that contains every urelement singleton, the inner model $U^\mathcal{I}$, which contains all urelements and sets whose kernel is in $\mathcal{I}$, is a model of ZFU$_R$ also. And $U^\mathcal{I}$ satisfies AC if $U$ does. So in the case of ZFCU$_\text{R}$, the Hartog number of a proper class can be any infinite cardinal.

For ZFCU$_\text{R}$ + Collection, the Hartog number of a proper class cannot be any limit cardinal by the argument you have given. But this is the only constraint: for every successor infinite cardinal $\kappa^+$, there can be models of ZFCU$_\text{R}$ + Collection where the Hartog number of the class of urelements is $\kappa^+$. This follows from Lemma 21 and Theorem 26 in my dissertation.

The result for ZFCU$_\text{R}$ CAN be extended to GBU$_\text{R}$ (in fact, KMU$_\text{R}$) with a global choice function. This is due to Felgner (see Theorem 108 in my dissertation). In class theory with urelements, it is important to distinguish different versions of second-order AC. For example, the existence of a global choice function doesn’t imply the universe can be well-ordered, as shown in Felgner’s model.

However, it is not known, for example, if KMU$_\text{R}$ + Collection + Global Choice (not Global Well-Ordering) is consistent with a proper class with Hartog number, say, $\aleph_1$.

Yes. If you start with infinitely many urelements and then take the sets whose kernel is finite (the kernel of a set is the set of urelements in its transitive closure), the resultant inner model will satisfy Replacement. A more generalized argument is included in my dissertation (https://arxiv.org/abs/2303.14274, Theorem 26).

Let ZFU$_\text{R}$ denote ZF set theory with urelements axiomatized with Replacement. In general, given a model $U$ of ZFU$_R$ an ideal $\mathcal{I}$ of the class of all urelements that contains every urelement singleton, the inner model $U^\mathcal{I}$, which contains all urelements and sets whose kernel is in $\mathcal{I}$, is a model of ZFU$_R$ also. And $U^\mathcal{I}$ satisfies AC if $U$ does. So in the case of ZFCU$_\text{R}$, the Hartogs number of a proper class can be any infinite cardinal.

For ZFCU$_\text{R}$ + Collection, the Hartogs number of a proper class cannot be any limit cardinal by the argument you have given. But this is the only constraint: for every successor infinite cardinal $\kappa^+$, there can be models of ZFCU$_\text{R}$ + Collection where the Hartogs number of the class of urelements is $\kappa^+$. This follows from Lemma 21 and Theorem 26 in my dissertation.

The result for ZFCU$_\text{R}$ can be extended to GBU$_\text{R}$ (in fact, KMU$_\text{R}$) with a global choice function. This is due to Felgner (see Theorem 108 in my dissertation). In class theory with urelements, it is important to distinguish different versions of second-order AC. For example, the existence of a global choice function doesn’t imply the universe can be well-ordered, as shown in Felgner’s model.

However, it is not known, for example, if KMU$_\text{R}$ + Collection + Global Choice (not Global Well-Ordering) is consistent with a proper class with Hartogs number, say, $\aleph_1$.

Yes. If you start with infinitely many urelements and then take the sets whose kernel is finite (the kernel of a set is the set of urelements in its transitive closure), the resultant inner model will satisfy Replacement. A more generalized argument is included in my dissertation (https://arxiv.org/abs/2303.14274, Theorem 26).

This result CAN be extended to GB (in fact, KM) with urelements with a global choice function. This is due to Felgner (see Theorem 108 in my dissertation). In class theory with urelements, it is important to distinguish different versions of second-order AC. For example, the existence of a global choice function doesn’t imply the universe can be well-ordered, as shown in Felgner’s model.

Yes. If you start with infinitely many urelements and then take the sets whose kernel is finite (the kernel of a set is the set of urelements in its transitive closure), the resultant inner model will satisfy Replacement. A more generalized argument is included in my dissertation (https://arxiv.org/abs/2303.14274, Theorem 26).

Let ZFU$_\text{R}$ denote the ZF set theory with urelements axiomatized with Replacement. In general, given a model $U$ of ZFU$_R$ an ideal $\mathcal{I}$ of the class of all urelements that contains every urelement singleton, the inner model $U^\mathcal{I}$, which contains all urelements and sets whose kernel is in $\mathcal{I}$, is a model of ZFU$_R$ also. And $U^\mathcal{I}$ satisfies AC if $U$ does. So in the case of ZFCU$_\text{R}$, the Hartog number of a proper class can be any infinite cardinal.

For ZFCU$_\text{R}$ + Collection, the Hartog number of a proper class cannot be any limit cardinal by the argument you have given. But this is the only constraint: for every successor infinite cardinal $\kappa^+$, there can be models of ZFCU$_\text{R}$ + Collection where the Hartog number of the class of urelements is $\kappa^+$. This follows from Lemma 21 and Theorem 26 in my dissertation.

The result for ZFCU$_\text{R}$ CAN be extended to GBU$_\text{R}$ (in fact, KMU$_\text{R}$) with a global choice function. This is due to Felgner (see Theorem 108 in my dissertation). In class theory with urelements, it is important to distinguish different versions of second-order AC. For example, the existence of a global choice function doesn’t imply the universe can be well-ordered, as shown in Felgner’s model.

However, it is not known, for example, if KMU$_\text{R}$ + Collection + Global Choice (not Global Well-Ordering) is consistent with a proper class with Hartog number, say, $\aleph_1$.