I heard from a reputable mathematician that ZFC proves that there is a minimal countably saturated real-closed field. I have several questions about this.

Is there a soft model-theoretic construction showing that there must be such a field? I can imagine an argument using the nice model-theoretic features of real-closed fields, such as quantifier-elimination etc. (And if not, what is the best argument? Formal power series were mentioned, so I guess something like Hahn series.)

Is the minimal countably saturated real-closed field also least, in the sense that it embeds into all other such fields?

Is it unique? (I would be amazed, but this would be very welcome as a canonical structure.)

Is the minimal countably saturated real-closed field simply $\text{No}_{\omega_1}$, that is, the surreal numbers born at any countable ordinal birthday? This field seems to be constructed in a minimal-like manner so as to be countably saturated, and so I would easily believe it is the one, if there is one. (Or perhaps it is one of several?)

I would appreciate any elucidation of these and related matters.

Meanwhile, I am aware that under CH, all smallest-size countably saturated real-closed fields are isomorphic, and furthermore, that this is equivalent to the continuum hypothesis. This fact was a central theme of my recent paper, How the continuum hypothesis could have been a fundamental axiom. So this question is mainly about the pure ZFC result, which is difficult only in the not CH case. In general, without CH, there are many non-isomorphic countably saturated real-closed fields of size continuum.