Generally, the structures that people study in every-day modern mathematics can be seen as defined on sets, so far as I know. (In the sense that they are collections of objects that don't give rise to russel's paradox).
E.g. the collection of real numbers, or of $5$-dimensional manifolds or of symmetry maps on the unit circle or whatever, are all sets.
However, I recall that sometimes structures cannot be assumed to be sets. E.g. we cannot speak of the Category of Categories apparently (unless we mean the category of small categories), since this would not be a set, and would give rise to russel's paradox (if I underatand correctly)
My question is: is it generally required for any mathematical theory that the domain of any arbitrary structure that satiafies it is a set, rather than generally a collection? E.g. do we require that a group not only satiafies the axioms, but implicitly require also that the domain of a particular group is a set?
No. For one thing a set can generally be defined without use of the words group, or collection, as being: a mathematical object, containing other objects, without repeat,or ordering. Order only comes into play in ordered-set related things. Repeats usually only come into play, when generalizing to multisets. To set-theory $2^4$ has 1 prime factor, to the theory of multisets it has 4 (as represented by the exponent), as it counts the multiplicities (repeats) as distinct.