I just meant that if $|A|=|\cup x|$, meaning they are equinumerous, then $A$ is equinumerous with $\cup x$, and so we can replace $x$ with $x'$ where $A=\cup x'$. So that part of your change didn't actually change anything. Meanwhile, the rest of my answer here does use that $\omega_1$ is well-orderable, so that all the cardinals below it are also well-orderable, and this is how I concluded that $x$ must be countable and that the elements of $x$ must be countable as well, and indeed finite for the hypersingular case.

I take same-cardinality to mean equinumerous, so we can assume $\omega_1=\cup x$ without loss.

But with your edit, the definition is now not well defined, since the property depends on $x$ and not just on $\cup x$. What I had meant was for you to say that a set $A$ is supersingular, if there is an $x$ for which $A=\cup x$ and so forth.

To match the familiar terminology, one should have the adjectives "supersingular", "ultrasingular" etc. apply to the set $\cup x$, not $x$. After all, $\aleph_\omega$ is singular, because it is the union of $x=\{\aleph_0,\aleph_1,\aleph_2,\ldots\}$, but we don't say that that set is singular. A cardinal is singular, when it is the size of a set that is the union of a small number of small sets.

I have now updated my answer to include this improved form of the argument.

Ah, that's very nice. So this rules it out entirely for ZF models.

Even with Gabe's suggestion, there will be a trivial example. Just take any ultrafilter on $\omega$, and use it to define an ultrafilter on Ord, giving measure one to $\omega$. Since this isn't really what you want, you should require uniformity, that is, that every bounded set of ordinals has measure zero.

It is also common to define the rank of a node $y$ in a well-founded relation to be $\rho(y)=\sup\{\rho(x)+1\mid x<y\}$, which amounts to the same thing as your levels.

To me, Mathias forcing is defined exactly as I had said in my comment above, with no reference to any filter at all. The modified forms of it, defined relative to a filter, came later and are to be seen as qualified forms of Mathias forcing.

I'd be happy to take a look. But I recognize that you and I have rather different approaches to studying the surreals.