My quesion is: What set theory are the mathematicians who are developing the theory of these numbers working in-or are they, in fact, working outside any of the standard set theories?. Each surreal number is a mapping of an ordinal number into the pair (+,-) so that the collection S of all these numbers is a proper class. Moreover S is a real closed (ordered) field containing sub-collections which are ordinally similar to the class of ordinal numbers and to the set of real numbers (in their usual order). Since S is densely ordered but not order-complete, there exists an order-complete ordered collection C (constructed from the Dedekind cuts of S), which contains a dense sub-collection that is ordinally similar to S. Now the elements of C are proper classes and if we are going to have theorems about sub-collections of C (such as closed intervals), then the underlying set theory (if any) must be one that allows some proper classes to be elements of collections.
As you said, each surreal number can be regarded as a set, but the collection of all of them is a proper class. The set-theoretic issues involved in "developing the theory of these numbers" are the same as those involved in developing the theory of sets. For most purposes, ZFC suffices, since particular proper classes can be handled as "virtual classes" (essentially, formulas with set parameters). If one really needs to quantify over proper classes (whether of sets or of surreal numbers), then a set-class theory like Morse-Kelley becomes appropriate. If even higher types are needed, then I would be inclined to drop this one-step-at-a-time approach and instead assume that there is an inaccessible cardinal $\kappa$ and that Conway was really working in the universe of sets of rank $<\kappa$.
Note that difficulties with proper classes had to be faced in the foundations of category theory, and several approaches have been developed, including Grothendieck's universes and Feferman's approach based on the reflection principle of ZFC. As far as I can see, these approaches can be adapted to deal with the analogous problems that arise in connection with surreal numbers.
Philip Ehrlich at Ohio University has written extensively on Conway's surreal numbers, and somewhere in his work he has the details for formalizing the theory of surreal numbers in NBG set theory. This should qualify as a "standard" set theory despite its use of classes as it's a conservative extension of ZFC, as mentioned above. His forthcoming paper for the Bulletin of Symbolic logic gives his paper
Absolutely saturated models. Fund. Math. 133 (1989), no. 1, 39–46.
as a reference for this formalization.
Why on Earth do you feel the need to coerce Conway's system into some form of Set Theory?
Please go and read the Appendix to Part Zero of On Numbers and Games, which is a manifesto for the Mathematician's Lib Movement.
Conway's games are already formulated in much the same way as set theories are, the only difference being that they have left- and right-handed members.
Why do you think that an official one-handed set theory is any more suitable as a foundation than Conway's system free-standing?
Anyway, if you have coerced it into your preferred set theory, it becomes useless for anybody else's. For example, Mike Shulman recently solved the problem of eliminating the double negations from every level, as part of the presentation of Conway Numbers in Homotopy Type Theory (in the final chapter of the book).
Garabed, I believe the class of surreals can be encoded by a class formula in ZFC.
Surreal numbers are particular kinds of Conway games, and each Conway game can be expressed as a well-founded rooted tree of bounded (i.e., non-class) size equipped with a labeling of the edges by symbols $L$ and $R$. (The nodes are positions in the game, with the root at initial position, and an edge from a node to a child is labeled $L$ or $R$ according to whether the child is a left or right option of the node.) So the full structure of a Conway game is fully specified by a set, and the class of Conway games can be given by a ZFC class formula. The relation $\lt$ on games, and the predicate that says a game is a number, are recursive and can be given by formulas in ZFC. Similarly, the equality predicate on numbers is recursive and given by a ZFC formula.
I don't have On Numbers and Games immediately to hand, but my memory is that Conway discusses these issues.