In model theory, if L is a first order language, by the definition of a L-structure $\mathcal{M}$ it is partly given by a non-empty set $M$ called the universe or domain of $\mathcal{M}$.
From where did we get this set? I know how I could define a first order set theory by giving the language L and then axioms of ZFC for example. Do we assume the existence of the set theory when reasoning about the models, interpretation, embeddings, etc.? What kind of set theory is it? What statements about this set theory are true and which or not?
In short: Is there any starting point, if yes, what is it?
Yes. Usually we do model theory in $\sf ZFC$, but sometimes we need to assume further assumptions (e.g. $\sf CH$ or large cardinals of some sort, etc.).
Thinking about it deeper, if the language is infinite, where does it live? If it is uncountable then we can't even fully express it in "plain logic" without appealing to some set theory.
I'm not sure what statements are "true about this set theory", because that would probably solve a lot of open questions in mathematics.