To see the exposure to FX risk and the difficulty for hedging, we assume constant interest rates and constant volatilities. Let $r_d$ and $r_f$ denote respectively the interest rates for USD and EUR. Moreover, let $X_t$ be the exchange rate at time $t$ from one unit USD to units of EUR. Finally, let $S_t$ be the price level of DAX at time $t$. We assume that, under the EUR risk-neutral measure $Q_f$,
\begin{align*}
\frac{dX_t}{X_t} &= (r_f-r_d) dt + \sigma_X dW_t\\
\frac{dS_t}{S_t} &= r_f dt + \sigma_S\big( \rho dW_t + \sqrt{1-\rho^2} dB_t\big),
\end{align*}
where $\{W_t, t \ge 0\}$ and $\{B_t, t \ge 0\}$ are two independent standard Brownian motions, and $\rho$ is the correlation.
Let $Q_d$ be the USD risk-neutral measure. Note that
\begin{align*}
\frac{dQ_d}{dQ_f}\big|_t = \frac{e^{r_dt} X_t}{e^{r_f t}X_0}.
\end{align*}
Therefore,
\begin{align*}
e^{-r_dT} E_{Q_d} \left(\frac{S_T}{S_0} \right) &=e^{-r_dT} E_{Q_f} \left(\frac{S_T}{S_0} \frac{e^{r_dT} X_T}{e^{r_f T}X_0}\right)\\
&= e^{-r_fT} E_{Q_f} \left(\frac{S_T X_T}{S_0X_0}\right) \tag{1}\\
&= e^{(r_f-r_d)T}E_{Q_f}\left(e^{-\frac{1}{2}\sigma_S^2 T +\sigma_T (\rho W_T + \sqrt{1-\rho^2} B_T) -\frac{1}{2}\sigma_X^2 T + \sigma_X W_T)} \right)\\
&=e^{(r_f-r_d)T}e^{\rho\sigma_S\sigma_X T}. \tag{2}
\end{align*}
Here, $\rho\sigma_S\sigma_X T$ is the quanto adjustment.
From $(1)$, we note that the payoff $S_T/S_0$ of the quanto forward, in USD, is equivalent to the payoff $S_TX_T/(S_0X_0)$ in EUR. Therefore, a quanto forward is exposed to FX risk. We also note that, though the FX rate does not explicitly appear in valuation formula $(2)$, both the FX risk factors $\rho$ and $\sigma_X$ are presented. Moreover, Formula $(2)$ is based on the constant volatility assumption, while in the local volatility framework, the volatility $\sigma_X$ will depend on the spot FX rate $X_0$. Regarding replication, we basically need to replicate the payoff $S_TX_T/(S_0X_0)$ in EUR, which does not appear an easy exercise.
EDIT: There is an interesting discussion of a similar product in Section 12.4.5 of the book Financial Risk Management.
