I need help with this definition from "Category Theory" at SEP:
According to the article, "This is the definition one finds in most textbooks of category theory."
Let's start with the first sentence. Does it mean that $Ob \subset C$?
If so, what else might be in $C$?
In the second sentence, are $X$ and $Y$ assumed to be elements of $Ob$?
Is a morphism from $X$ to $Y$ the same as a function mapping elements of $X$ to elements of $Y$?
The definition of a category is twofold. If we want to say that something is a category you need to specify the objects your category consists of together with the morphisms between them. So in a way, one can identify a category $\mathbf{C}$ with its "class" of objects $\textbf{Ob}$. However by doing that you are not specifying the morphisms, so people usually write "let $X \in \mathbf{C}$" instead of "let $X \in \textbf{Ob}$" when it is clear which category we are working with.
This means that no, $\textbf{Ob}$ is not a subset of $\mathbf{C}$. Object-wise we have that $\textbf{Ob}$ is $\mathbf{C}$, so there is no more stuff outside $\mathbf{Ob}$.
Next, a category consists of, for each pair of objects $X$ and $Y$ in $\mathbf{Ob}$, a set $\textbf{Hom}(X,Y)$, so this means that yes, given a morphism $f \colon X \to Y$ in your category, both the target and the source are objects in your category.
Your fourth question has a negative answer however. Although it is true that in the first examples of categories that come to mind the objects are "sets plus extra information" (think of the category of sets, groups, abelian groups, topological spaces, etcetera), it is not true in general that an object in an arbitrary category has the underlying structure of a set. Therefore it doesn't make sense to talk about elements of an object at all.
Maybe this is better explained with an example. Let $\mathbf{C} = \mathbf{1}$ denote the category whose only object is the symbol $\star$ and the identity morphism $\text{id}_{\star}$ is the only morphism. This is a category, but notice that $\star$ is not a set and hence the (unique) morphism in this category can't be thought of as function mapping elements of $\star$ to itself.
$$ \star \to \bullet$$ is another category, where I've omitted the identity arrows for $\star$ and $\bullet$, provided that the morphism $\star \to \bullet$ respects the identity axiom. Again the only morphism that is not the identity is not a function between sets, since $\star$ and $\bullet$ might not be sets to begin with.
Finally, I would like to address part of the set-theoretical difficulties that appear inevitably when dealing with categories. We want to be able to work with categories such as $\textbf{Sets}$, whose objects are sets and the set of morphisms $\textbf{Hom}(X,Y)$ for any two sets $X$ and $Y$ are functions between sets. Then, what "thing" is $\textbf{Ob}$ in this category? It can't be a set since Russell's paradox shows the inconsistency of allowing a "set of all sets". One way to get rid of this difficulty is to allow classes, and to work in an axiomatic set theory that has this notion, so that one says "$\textbf{Ob}$ is a class".
Another possibility which I personally find cleaner is to fix what is known as a Grothendieck Universe $\mathcal{U}$ and talk about the category of $\mathcal{U}$-sets, $\mathcal{U}$-groups, etcetera. Anyway, this is more a matter of preference, I'd say.
Hope this helps!