I have found two different references which answer both my questions in the affirmative. The first is Zariski and Samuel's 'Commutative Algebra I' (Chapter V, p. 317, Thm. 34) and the other is Neukirch's 'Algebraic Number Theory' (Chapter I, p. 47-48, Prop. 8.3). It seems that the only remaining interesting question is: What exactly happens when the field extension is not separable?
Zariski and Samuel describe a theorem of Kummer which applies to integrally closed domains (more general than Dedekind domains), but restricts to the case where the larger ring $R'$ is of the form $R[y]$, while in Dedekind's theorem, $R[y]$ might be a finite-index subgroup of $R'$. In Kummer's theorem, prime ideals are replaced with maximal ideals (in Dedekind domains, those two notions essentially coincide).
Neukirch describes a theorem which applies to Dedekind domains, and gives exactly Dedekind's Theorem when one works with rings of integers.
Dedekind's theorem is a slight strengthening of Kummer's theorem, specialized to the case of number fields. It doesn't follow directly from Kummer's theorem, but it does follow from the theorem described by Neukirch.
Kummer's theorem is the following:
Let $R$ be an integrally closed domain, $K$ its quotient field, $K'$ a
finite algebraic extension of $K$, $R'$ the integral closure of $R$ in
$K'$. We suppose that there exists an element $y$ of $R'$ such that
$R'=R+Ry+\cdots + Ry^{n-1}$ ($n=[K':K]$) ($y$ is then a primitive
element of $K$' over $K$). Let $F(Y)$ be the minimal polynomial of $y$
over $K$. ($F(Y)$ necessarily has its coefficients in $R$.) Let
$\mathfrak{p}$ be a maximal ideal in $R$; for every polynomial $G(X)$
over $R$, we denote by $\overline{G}(X)$ the polynomial over
$R/\mathfrak{p}$ whose coefficients are the $\mathfrak{p}$-residues of
the corresponding coefficients of $G$. Let
$\overline{F}(X)=\prod_{i=1}^{g} (f_i(X))^{e(i)}$ be the factorization
of $\overline{F}(X)$ into distinct irreducible factors $f_i(X)$ over
$R/\mathfrak{p}$; for $i=1,\cdots,g$ we denote by $F_i(X)$ a
polynomial over $R$ such that $\overline{F_i}(X) = f_i(X)$. Then the
ring $R'$ has exactly $g$ maximal ideals $\mathfrak{P}_i$ which lie
over $\mathfrak{p}$, and we have $$\mathfrak{P}_i = R'\mathfrak{p}+R'F_i(y).$$
Furthermore we have $$R'\mathfrak{p} =\mathfrak{D}_1 \cap \mathfrak{D}_2 \cdots \cap \mathfrak{D}_g = \mathfrak{D}_1 \cdot\mathfrak{D}_2 \cdots \cdot \mathfrak{D}_g,$$ where
$\mathfrak{D}_i = R'\mathfrak{p} + R'(F_i(y))^{e(i)}$.
I have found Neukirch's theorem here. $\mathcal{o}$ stands for a Dedekind domain, $K$ for its fraction field, $\mathcal{O}$ for the integral closure of $\mathcal{o}$ in $L$, a finite separable extension of $K$:
