Smallest $s$-component in mincut
I believe that the answer is yes: any min-cut constructed in this way from a max-flow will also have minimum possible cardinality, among all possible min-cuts.

There can be multiple min-cuts, all with the same cost.  They form a lattice structure: the intersection of two min-cuts is another min-cut, and the union of two min-cuts is another min-cut.  You can identify a "smallest" element in this lattice by taking the intersection of all min-cuts; this will be another min-cut, and it will have the smallest cardinality of all min-cuts.

As I understand it, it is possible to prove that the min-cut obtained from a max-flow is always this "smallest" min-cut.  Or, to put it another way, if you think of the source $s$ on the left and the sink $t$ on the right, then any min-cut obtained from a $s,t$-max-flow will be always be a "leftmost" cut.  Also, it will follow that any other min-cut will be a superset of this cut found by max-flow, exactly as you conjectured.

For references to these results, and other related material, see the following questions (note: you may need to double-check some of the claims yourself, as I have not personally verified them):

Does Ford-Fulkerson always produce the left-most min-cut

https://stackoverflow.com/a/8101250/781723

https://stackoverflow.com/q/26696312/781723

https://stackoverflow.com/q/9210755/781723

https://stackoverflow.com/q/41964288/781723