What role does the "dual Coxeter number" play in Lie theory (and should it be called the "Kac number")?
The dual Coxeter number comes up naturally as a normalization factor for invariant bilinear forms on the Lie algebra: according to Kac's book which you quote, $2h^{\vee}$ is the ratio between the Killing form and the "minimal" bilnear form (the trace form for $sl_n$), which has the property that the square of the length of the maximal root is 2.
This minimal form corresponds to the minimal affine Kac-Moody group corresponding to the Lie algebra, or equivalently to the minimal line bundle on the affine Grassmannian or the moduli spaces of G-bundles on curves (the generator of the Picard group). As a result, the $-2h^\vee$-th power of the basic ample line bundle on the Grassmannian or moduli space of bundles (which is associated to the level given by the Killing form) ends up being identified with the canonical line bundle, and in particular the $h^\vee$th power is a square-root of the canonical bundle, or spin structure. (This is analogous to the role of $\rho$ for the finite flag variety.) Thus the critical level arises naturally geometrically -- it corresponds to half-forms on the Grassmannian/moduli spaces. The basic yoga of quantization (or of unitary/normalized induction of representations) tells us that classical symmetries are "shifted" by half-forms - cf $\rho$-shifts in representation theory. Likewise the critical shift for affine algebras.. for example the Feigin-Frenkel theorem is the analogue of the Harish-Chandra isomorphism: the center of the enveloping algebra at critical level (rather than level 0 as one might naively guess, ignoring half-form twists) is isomorphic to the algebra of invariant polynomials on the (dual of the) Lie algebra. (This can be said more canonically keeping track of symmetries of change of variable, magic word being "opers", but let's ignore that).
One can say all this very naturally algebraically (without resorting to geometry) -- $\rho$ can be described as the square root of the modular character of the Borel subalgebra (up to sign or something, not being very careful here). The critical level has a similar description in terms of the positive half (Taylor series part) of the Kac-Moody algebra - if you try to define the modular character of this half you are quickly led to semiinfinite determinants etc, ie to the previous geometric story, and so one can assert that the critical level "is" half the modular character of the positive loop subalgebra.
I think part of the question here is "why is this thing called the dual Coxeter number? It looks pretty different, so why don't we just give it a different name?" I think the case is made in Kac's book that dual Coxeter number is the right name.
The Coxeter number for $\mathfrak{g}$ is the sum of the labels in the Dynkin diagram for the untwisted affine algebra corresponding to $\mathfrak{g}$. These labels are the coefficients of a minimal integer linear dependence among the columns of the affine Cartan matrix, which seems fairly intrinsic, so I think this is a reasonable definition. I won't try to explain why it is equivalent to more standard definitions. The dual Coxeter number is then the sum of the labels in the dual affine Dynkin diagram. See Kac, section 6.1 for these definitons.
I think what is confusing is that "dual" and "affine" do not commute. For instance, the dual of the affine diagram of type $B_\ell^{(1)}$ is the twisted affine Dynkin diagram of type $A_{2\ell-1}^{(2)}$.
Let $\mathfrak{g}$ and $\mathfrak{h}$ be semisimple Lie algebras corresponding to connected simply connected compact groups $G,H$. Any map $\mathfrak{g} \to \mathfrak{h}$ has a Dynkin index, which is the induced map $\mathrm{H}^4(BH) \to \mathrm{H}^4(BG)$. When $\mathfrak{g}$ and $\mathfrak{h}$ are simple, $\mathrm{H}^4(BH)$ and $\mathrm{H}^4(BG)$ are both isomorphic to $\mathbb Z$ (and this isomorphism can be chosen canonically by using the generator which maps to a positive-definite element under $\mathrm{H}^4(BG) \to \mathrm{H}^4(BG) \otimes \mathbb{R} \cong \mathrm{Sym}^2(\mathfrak{g})^W$), and the Dynkin index is then just a number.
The dual Coxeter number of $\mathfrak{g}$ arises as the Dynkin index of the map $\mathrm{adj} : \mathfrak{g} \to \mathfrak{so}(\dim\mathfrak{g})$. (Almost. When $\dim \mathfrak g \leq 4$, this fails. What is correct is to look at the stablized adjoint map $\mathfrak{g} \to \mathfrak{so}(\infty) = \varinjlim \mathfrak{so}(n)$.)
What's called "the Dynkin index of a representation" $V$ is the Dynkin index of the map $\mathfrak{g} \to \mathrm{sl}(\dim V)$. Note that $\mathfrak{so}(n) \to \mathfrak{sl}(n)$ has Dynkin index $2$ when $n \geq 5$, explaining the factor of two in the usual formulas about $2h^\vee$ and Killing forms.