Yes this is true, formally this follows by the envelope theorem. In an abstract and very smooth setting, the envelope theorem says that for an objective functional depending on a parameter $t$
$$
F(t)=\max\limits_z f(t,z),
$$
then the derivative of the optimal value can be computed as
$$
\frac{dF}{dt}(t)=\partial_t f(t,z_t)
\qquad
\mbox{for any smooth selection of a maximizer }z_t \mbox{ of }F(t).
$$
This can be seen easily: for any such choice of a maximizer, just apply a chain rule and use the optimality condition of $z_t$ in the maximization problem for fixed $t$:
$$
F'(t)=\frac d{dt}f(t,z_t)=\partial_t f(t,z_t)+\underbrace{\partial_zf(t,z_t)}_{=0}\frac {dz_t}{dt}.
$$
This means, roughly speaking, that one can simply forget that the minimizer varies, only the variation of the functional matter.
In your specific context, you are trying to differentiate (w.r.t $\rho$) the optimal value of the optimization problem given by the Kantorovich dual formulation
$$
W^2(\rho,\eta)
=F_\eta(\rho)
=\max\limits_\phi f_\eta(\rho,\phi)
=\max\limits_\phi \left\{\int \rho\phi+\int\eta\phi^c\right\}
$$
(here $\eta$ is fixed once and for all, I'm mimicking my $F,f$ notations above to give some perspective and I hope the notation is sufficiently self-explanatory).
Although the Kantorovich potential $\phi$ from $\rho$ to $\eta$ (the optimizer) varies when $\rho$ varies, the envelope theorem strongly suggests that you can actually argue as it did not vary at leading order (same for its $c$-transform $\phi^c$), and one can simply differentiate the functional w.r.t. the varying "parameter" $\rho$. Since the Kantorovich functional is linear in $\rho$, the conclusion is indeed that the first variation is given by $ \frac{\partial f_\eta}{\partial_\rho}(\rho,\phi)=\phi$.
Of course various subtle problems may arise owing essentially to the infinite-dimensional setting and functional-analytic details, but this is the rough idea.
For a completely rigorous statement and proof I can recommend Filippo Santambrogio's book [1], in particular chapter 7 and Proposition 7.17
[1] Santambrogio, Filippo. "Optimal transport for applied mathematicians." Birkäuser, NY 55.58-63 (2015): 94.
$\newcommand{\ep}{\varepsilon}\newcommand\R{\mathbb R}$Yes, the minimizer $\mu$ is absolutely continuous (w.r. to the Lebesgue measure $|\cdot|$).
Indeed, you showed that
\begin{equation*}
F(\rho_n)\to m:=\inf_\rho F(\rho) \tag{-1}
\end{equation*}
and
\begin{equation*}
\mu_{\rho_n}\to\mu \tag{-0.5}
\end{equation*}
weakly for some sequence $(\rho_n)$ of probability densities and some probability measure $\mu$, where
\begin{equation*}
F(\rho):=\int\rho\ln\rho\,dx+W_2^2(\rho,\eta) \tag{0}
\end{equation*}
and $\mu_\rho(dx):=\rho(x)\,dx$.
Take any set $E\subseteq\R$ with $|E|=0$. We have to show that then $\mu(E)=0$.
Take any real $\ep>0$. By the regularity of the Lebesgue measure, there is an open set $G_\ep\subset\R$ such that
\begin{equation*}
\text{$E\subseteq G_\ep$ and $|G_\ep|<\ep$.} \tag{0.5}
\end{equation*}
By (-0.5) and the Portmanteau theorem,
\begin{equation*}
\mu(G_\ep)\le\liminf_n\mu_{\rho_n}(G_\ep). \tag{1}
\end{equation*}
Next, for each real $a>1$,
\begin{equation*}
\mu_{\rho_n}(G_\ep)=K_n+L_n, \tag{2}
\end{equation*}
where
\begin{equation*}
K_n:=\int_{G_\ep\cap[\rho_n\le a]}\rho_n\,dx,\quad L_n:=\int_{G_\ep\cap[\rho_n>a]}\rho_n\,dx,
\end{equation*}
$[\rho_n\le a]:=\rho_n^{-1}((-\infty,a])$, $[\rho_n>a]:=\rho_n^{-1}((a,\infty))$.
Further,
\begin{equation*}
K_n\le a|G_\ep|<a\ep \tag{3}
\end{equation*}
by (0.5), and
\begin{equation*}
L_n\le \int_{[\rho_n>a]}\rho_n\,dx
\le\frac1{\ln a}\int \rho_n\ln\rho_n\,dx\le \frac{m+1}{\ln a} \tag{4}
\end{equation*}
for all large enough $n$, by (-1) and (0).
By (0.5), (1), (2), (3), (4),
\begin{equation*}
\mu(E)\le\mu(G_\ep)\le a\ep+\frac{m+1}{\ln a},
\end{equation*}
for all real $\ep>0$ and all real $a>1$. Letting now $\ep\downarrow0$ and then $a\to\infty$, we get $\mu(E)=0$, as desired.
Best Answer
You can find this in Villani's "small book", Theorem 8.13 in [Villani, C. (2003). Topics in optimal transportation (Vol. 58). American Mathematical Soc.]
I can also recommend looking at Filippo Santanbrogio's more recent and applied book [Santambrogio, F. (2015). Optimal transport for applied mathematicians. Birkäuser, NY, 55(58-63), 94.] in particular chapter 7 (a more general version of your statement can be found in section 7.2.2)