I apologize in advance if anything I say in the following is incorrect and I appreciate any corrections if anything is incorrect. My knowledge of set theory and logic is extremely limited, however, I try to read something about those topics on my own as I feel like it adds to "everyday mathematics".
In ZFC the notions of set and set membership are not defined but rather described axiomatically by the ZFC axioms. Thus, as far as I understand, sets are just that, whatever that is and by the axioms one can derive statements about those objects.
In naive set theory any collection of objects (where objects is a primitive notion as well; anything is an object) is viewed as a set intuitively representing the notion of a collection of abstract objects. Without precise axioms however, this leads to paradoxes such as Russels Paradox.
I heard of a philosophical view, which views mathematics as a game or a formal manipulation of strings, called formalism. However, if one motivates the axioms of ZFC a bit more platonistic, namely by formalizing objects that occur in the real world such as numbers, one would probably still view a set as a formalization of a collection of objects under certain restrictions, which are given by the axioms. In this case, I could view a statement such as $s \in S$ more intuitively as "$s$ names an object which is contained in the collection called $S$, whatever that object may be."
Question 0) I have no knowledge about interpretation/semantics of formal logical sentences and how it works exactly. Are there good references to study those? I am not sure how long it would take, but I would find it interesting to read about in my free time. It may have been incorrect, but I myself have always thought that mathematical statements always have a certain interpretation, which is why I think this might be an interesting topic for me to read about. I think that strictly speaking statements are formal at first without interpretation, but can be interpreted, right?
Question 1) Are sets in ZFC also interpreted as collections of objects or does one not use an interpretation at all? Since there are probably a wide variety of views, I should reformulate: Is it valid to interpret sets as a collection of objects and objects being a member of the set if and only if they satisfy the "$\in$" relation, or is it invalid?
Question 2) Is this view used by (some) professional mathematicians or is this view of a set as a collection mostly abandoned after encountering that naive set theory leads to contradictions? I myself am not sure which view is adopted in lectures for example, since only in introductory lectures it is explicitly mentioned that the naive view is adopted. However, I find it way easier to deal with the concepts by viewing sets as collections and elements as objects in the set. Therefore I wanted to ask whether it is valid to keep this view, or if this view is mostly abandoned.
A thought: I could view maths from a formalist point of view, but I myself find it more motivating and realistic to have a more platonic view. Wouldn't it be even more surprising that maths can be used in the real world if one adopts a purely formalistic view?
Try the fine book by Michael Potter, Set Theory and its Philosophy (OUP)
Or for other, even more introductory, suggestions see the main chapter on set theory in Beginning Mathematical Logic: A Study Guide (www.logicmatters.net/tyl)