I believe the following gives a positive answer to your first question, modulo large cardinals:

Let $\kappa$ be measurable, and consider a Prikry-generic extension of the universe $V[G]$. Look at $\mathcal{M}=(V_{\kappa+1})^{V[G]}$. We can view this as a model of $MK^-$ by taking the proper classes to be exactly those elements of $\mathcal{M}$ of rank $<\kappa$. In particular, the generic $G$ - which is a cofinal map from $\omega$ to $\kappa$ - is a "proper class" in the sense of $\mathcal{M}$, yet equinumerous with the $\mathcal{M}$-set $\omega$.

I strongly suspect that the measurable is unnecessary here - I'm just too lazy to think up the details for a ZFC-example (maybe something closer to Namba forcing?). The reason I'm using Prikry here is because Prikry forcing doesn't add any bounded-rank subsets of $\kappa$, which gives a fast (but probably unnecessary) proof that $\mathcal{M}$ is in fact a model of $MK^-$.

I believe the following gives a positive answer to your first question, modulo large cardinals:

Let $\kappa$ be measurable, and consider a Prikry-generic extension of the universe $V[G]$. Look at $\mathcal{M}=(V_{\kappa+1})^{V[G]}$. We can view this as a model of $MK^-$ by taking the proper classes to be exactly those elements of $\mathcal{M}$ of rank $\kappa$. In particular, the generic $G$ - which is a cofinal map from $\omega$ to $\kappa$ - is a "proper class" in the sense of $\mathcal{M}$, yet equinumerous with the $\mathcal{M}$-set $\omega$.

I strongly suspect that the measurable is unnecessary here - I'm just too lazy to think up the details for a ZFC-example (maybe something closer to Namba forcing?). The reason I'm using Prikry here is because Prikry forcing doesn't add any bounded-rank subsets of $\kappa$, which gives a fast (but probably unnecessary) proof that $\mathcal{M}$ is in fact a model of $MK^-$.

I believe the following gives a positive answer to your first question, modulo large cardinals:

Let $\kappa$ be measurable, and consider a Prikry-generic extension of the universe $V[G]$. Look at $\mathcal{M}=(V_{\kappa+1})^{V[G]}$. We can view this as a model of $MK^-$ by taking the proper classes to be exactly those elements of $\mathcal{M}$ of rank $<\kappa$. In particular, the generic $G$ - which is a cofinal map from $\omega$ to $\kappa$ - is a "proper class" in the sense of $\mathcal{M}$, yet equinumerous with the $\mathcal{M}$-set $\omega$.

I strongly suspect that the measurable is unnecessary here - I'm just too lazy to think up the details for a ZFC-example (maybe something closer to Namba forcing?).

I believe the following gives a positive answer to your first question, modulo large cardinals:

Let $\kappa$ be measurable, and consider a Prikry-generic extension of the universe $V[G]$. Look at $\mathcal{M}=(V_{\kappa+1})^{V[G]}$. We can view this as a model of $MK^-$ by taking the proper classes to be exactly those elements of $\mathcal{M}$ of rank $<\kappa$. In particular, the generic $G$ - which is a cofinal map from $\omega$ to $\kappa$ - is a "proper class" in the sense of $\mathcal{M}$, yet equinumerous with the $\mathcal{M}$-set $\omega$.

I strongly suspect that the measurable is unnecessary here - I'm just too lazy to think up the details for a ZFC-example (maybe something closer to Namba forcing?). The reason I'm using Prikry here is because Prikry forcing doesn't add any bounded-rank subsets of $\kappa$, which gives a fast (but probably unnecessary) proof that $\mathcal{M}$ is in fact a model of $MK^-$.