On page 183 of  “Fields of Surreal Numbers and Exponentiation”, Fundamenta Mathematica 167 (2001), pp. 173-188, Lou van den Dries and Philip Ehrlich prove the following result that I believe provides an answer to the author’s queries. The result is an improvement of a result the author attributes to Norman Alling, which was established independently by Ehrlich in “An Alternative Construction of Conway’s Ordered Field No,” Algebra Universalis 25 (1988), pp. 7-16. Ehrlich’s proof actually slightly predates Alling’s’s, but it appeared later. 

Proposition. Let  $\lambda$ be an epsilon number $\le On$ and let  $\tau  =1/\omega $. Then

(i)  $${\bf{No}}\left( \lambda  \right) = \bigcup\nolimits_\mu  \mathbb{R}{\left( {\left( {\tau ^{{\bf{No}}\left( \mu  \right)} } \right)} \right)_\lambda  } $$ where  $\mu$ ranges over the additively indecomposable ordinals $ < \lambda $;

(ii)  ${\bf{No}}\left( \lambda  \right)$ is real-closed;

(iii)  $${\bf{No}}\left( \lambda  \right) = \mathbb{R}\left( {\left( {\tau ^{{\bf{No}}\left( \lambda  \right)} } \right)} \right)_\lambda  $$ if and only if  $\lambda $ is a regular cardinal;

(vi) For all  $y \in {\bf{No}}$,   $y \in {\bf{No}}\left( \lambda  \right)
$ if and only if  $\omega ^y  \in {\bf{No}}\left( \lambda  \right)$.