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An answer to the first question is: Take a model $\mathcal{M}$ of $\text{ZF +GCH}$, the stage $V_{w+w+1}$ of $\mathcal{M}$ will satisfy all axioms of $\text{MK}^-$ pluse the first two conditions of the first question: the stage $V_{w+w}$ would interpret the class $V$ of all sets of $\text{MK}^-$, a set of $\text{MK}^-$ is intepreted as an element of $V_{w+w}$, while a proper class of $\text{MK}^-$ is a subset of $V_{w+w}$ that is not an element of $V_{w+w}$; now we have $|V_{w+w}|=\aleph_w$, and since the Generalized Continuum Hypothesis "$\text{GCH}$" is equivalent to $\beth_\alpha=\aleph_\alpha$ for every infinite ordinal $\alpha$, then every cardinality less than the cardinality of $V_{w+w}$ would be either $n$ or $\aleph_{w+n}$ for a natural $n$, all of which are cardinalities of elements of $V_{w+w}$, and thus cardinalities of sets, that a proper class exists that is strictly subnumerous to $V$ is witnessed by the set of all $V_i$ stages with $i<w+w$, since it is a subset of $V_{w+w}$ and it is not an element of $V_{w+w}$ and its cardinality is $\aleph_0 <\aleph_w$. The rest of axioms of $\text{MK}^-$ (Extensionality, Class comprehension axioms, Pairing, Power, Infinity and Subsets) are interpreted straightforwardly.