Is a subobject classifier logically equivalent to set-inclusion? Can one subsume the notion of set-inclusion $\subseteq$ and $\subset$ with the notion of a subobject classifier, expressed via an injective morphism $\hookrightarrow$? Specifically: Are the expressions $Gal(K/k) \hookrightarrow Aut(K)$ and $Gal(K/k) \subseteq Aut(K)$ logically equivalent? If not, how are they different?

As Zhen Lin noted in his comment, subobject classifiers have nothing to do with the matter at hand.

Having said that: being a subobjet is not logically equivalent to being a subset. Not even in the category of sets and functions.

More exactly, in **Set** , $A \subseteq B$ implies that $A$ is a subobject of $B$ (the mono in question being the injection), but not vice-versa. Example: set {a,b,c} is a subobject of set {1,2,3,4}, but not a subset.

If you want to use categorical language and still use the concept of subset of a set $S$, you can use the category $\mathcal{P}(S)$ (powerset of $S$) whose objects are subsets of $S$ and morphisms are injections between the subsets (where present, of course). This category is a poset (at most one morphism between objects), it has initial and terminal objects (empty set and $S$) and it has categorical products and coproducts (intersection and union of subsets).