gr.group-theoryinfinite-combinatorics
To clarify the terms in the question above:
The symmetric group Sym($\Omega$) on a set $\Omega$ consists of all bijections from $\Omega$ to $\Omega$ under composition of functions. A generating set $X \subseteq \Omega$ is minimal if no proper subset of $X$ generates Sym($\Omega$).
This might be a difficult question, but perhaps the answer is known already?
This is an answer for the abelian case only.
For a finite abelian group (let us exclude the trivial one) there are two standard ways to decompose it as a direct sum of cyclic groups. One into cyclic groups of order $n_1,\dots,n_r$ (not $1$) such that $n_i \mid n_{i+1}$ and the other one into cyclic groups of order $m_1, \dots, m_s$ such that each $m_i$ is a prime power (not $1$). [Under each of the assumptions 'divisibility' and 'prime power' the repective decomposition is unique, in the latter case of course up to the ordering.]
The parameter $r$ is sometimes called the rank and the parameter $s$ the total rank of the group (although terminology here is not completely uniform).
Now, it is known that the rank is the minimal cardinality of a generating set, in the sense that there does not exists a set of a smaller cardinality that generates the group. And, that the total rank is the maximal cardinality of a minimal generating set, that is there exists a generating set of cardinailty $s$ such that no subset of this set generates the group.
Thus, the cardinality of all minimal/irredundant generating set is uniquely determined if and only if the rank equals the total rank. This is the case if and only if the group is an (abelian) $p$-groups.
Note that for a non-$p$-group on can first consider the first decomposition into cycylic groups, and then decompose each cyclic component as the sum of cyclic groups of prime power order (more or less Chines Remainder Theorem); so only if all the orders in the first decomposition are also prime powers (and thus necessarily powers of the same prime) does one not get a different value for rank and total rank.
The surreal numbers exhibit much stronger universal properties than you have mentioned, for they also exhibit very strong homogeneity and saturation properties. For example, every automorphism of a set-sized elementary substructure of the surreals extends to an automorphism of the entire surreal numbers, and every set-sized type over the surreals that is consistent with the theory of the surreal numbers is realized in the surreal numbers. That is, any first-order property that could be true about an object as it relates to some surreal numbers, which is consistent with the theory of the surreal numbers, is already true about some surreal number.
For any complete first-order theory $T$, one can consider the concept of a monster model of the theory. This is a model $\mathcal{M}$ of $T$, such that first, every other set-sized model of $T$ embeds as an elementary substructure of $\mathcal{M}$--not merely as a substructure, but as a substructure in which the truth of any first-order statement has the same truth value in the substructure as in big model--and second, such that every automorphism of a set-sized substructure of $\mathcal{M}$ extends to an automorphism of $\mathcal{M}$.
One may also use approximations to these proper class monster models by considering extremely large set-sized models, of some size $\kappa$, such that the embedding and homogeneity properties hold with respect to substructures of size less than $\kappa$.
Every consistent complete first order theory $T$ has such monster models, and they are used pervasively in model theory. Model theorists find it convenient, when considering types over and extensions of a fixed model, to work inside a fixed monster model, considering only extensions that arise as submodels of the fixed monster model.
Your topic also has a strong affinity with the concept of the Fraïssé limit of a collection of finitely-generated (or $\kappa$-generated) structures, which one aims to be the age of the limit structure, the collection of finitely-generated ($\kappa$-generated) substructures of the limit structure. Fraïssé limits are often built to exhibit the same saturation and homogeneity properties of the surreal numbers. As a linear order, the surreal numbers are the Fraïssé limit of the collection of all set-sized linear orders. And I believe that one can also put the field structure in here.
Update. When one has the global axiom of choice, the set-homogeneous property of the generalized Fraïssé limit allows one to establish universality for proper class structures. Basically, using global AC one realizes a given proper class structure as a union of a tower of set structures, and gradually maps them into the homogeneous structure. The homogeneity property is exactly what you need to keep extending the embedding, and so one gets the whole proper class structure mapping in. This kind of argument, I believe, shows that what you want to consider is homogeneity rather than merely the universal property itself. (This argument is an analogue of the idea that when one has homogeneity for countable substructures, then one gets universality for structures of size $\aleph_1$.)
Regarding your specific construction, here is a simpler way to undertake the same idea, which avoids the need for the equivalence relation: Let $G$ be the proper class of all fixed-point-free permutations of a set. This class supports a natural group operation, which is to compose them, regarding elements outside the domain as fixed by the permutation, and then cast out any newly-created fixed points. The identity element of $G$ is the empty function, which is really a stand-in for the identity function on the universal class.
It is clear that every group finds an isomorphic copy inside $G$, without using the axiom of choice, since every group is naturally isomorphic to a group of permutations, and these are naturally embedded into $G$, simply by casting out fixed-points. This does not use the axiom of choice.
The class group $G$ is a natural presentation of the set-support symmetric group $\text{Sym}_{\text{set}}(V)$ of the set-theoretic universe $V$, the class of all permutations of $V$ having set support. Any such permutation of $V$ is represented in $G$ by restricting to the non-fixed-points.
Note finally that in the case of the surreal numbers, one doesn't need the axiom of choice in order to construct the surreal numbers, and one can get many universal properties for well-orderable set structures and in general from the axiom of choice for all set-sized structures. But to get the universal property that you mention for class-sized structures, mere AC is not enough, for one needs the global axiom of choice, which is the assertion that there is a proper class well-ordering of the universe. This does not follow from AC, for there are models of Gödel-Bernays set theory GB that have AC, but not global choice. But global AC is sufficient to carry out the embedding of any class linear order into No. A similar phenomenon arises with many other class-sized class-homogeneous set-saturated models, which require global AC to get the universality for class structures.
Best Answer
I think it follows from Theorem 1.1 of "Subgroups of Infinite Symmetric Groups" by Macpherson and Neumann (J. London Math. Soc. (1990) s2-42 (1): 64-84) that there is no minimal generating set of $S(\Omega)$for infinite $\Omega$.
The theorem states that any chain of proper subgroups of $S(\Omega)$ whose union is $S(\Omega)$ must have cardinality strictly greater than $|\Omega|$.
Now suppose $X$ is a minimal generating set. Let $C=\{x_0,x_1,\dots\}$ be a countable subset of $X$. If $$H_i=\langle X\setminus C,x_0,\dots,x_i\rangle$$ for $i\in\mathbb{N}$, then $H_0<H_1<\dots$ is a countable chain of proper subgroups whose union is $S(\Omega)$, contradicting the theorem.
(Note: There's a paper of Bigelow pointing out some unstated set-theoretic assumptions in Macpherson and Neumann's paper, but I don't think that affects the theorem I mention.)