Is there a standard form for packaging up slice categories into a 2-category? For a category $A$ the slice categories are $A/a$.
Also, for any $f:a\rightarrow b$ we get a pullback functor $f^*:A/b\rightarrow A/a$. (This is only a functor if the objects in the slice categories are isomorphism classes).
Is there a standard means of packaging all these slices into a single category $\bar{A}$, which presumably makes a 2-category, and also has an embedding $A^{op}\rightarrow \bar{A}$? Which sort-of looks like some kind of Yoneda embedding.
 A: This is the theory of fibred categories (= Grothendieck fibrations). Given a category $\mathcal{C}$ with pullbacks, the fundamental fibration is the codomain functor $[\mathbb{2}, \mathcal{C}] \to \mathcal{C}$. It is easy to check that it is always a Grothendieck opfibration, and when $\mathcal{C}$ has pullbacks, it is also a Grothendieck fibration. The fibre of the fundamental fibration over an object $A$ is, of course, the slice category $\mathcal{C}_{/ A}$.
Alternatively, one could use the theory of indexed categories. As you say, there is an evident pseudofunctor $\mathcal{C}^\mathrm{op} \to \mathfrak{Cat}$ given by $A \mapsto \mathcal{C}_{/ A}$. This is called the self-indexing of $\mathcal{C}$. The category $[\mathbb{2}, \mathcal{C}]$ is equivalent to the Grothendieck construction of this pseudofunctor.
There isn't really a Yoneda embedding here, but if you really want to, you can view $A \mapsto \mathcal{C}_{/ A}$ as a Yoneda embedding: the projection functor $\mathcal{C}_{/ A} \to \mathcal{C}$ is a discrete fibration and corresponds to the presheaf $\mathcal{C}(-, A)$.
