Let $\{X_t\}$ be a stochastic process (say $t\in[\,0,T\,]$), where the random variables $X_t$ are supposed to be iid normal distributed with mean $= 0$, and variance $= 1$.
Consider the random variable
$$S_T=\int_0^T t^2 \,X_t \,dt$$
Obviously the expectation value $S_T$ is equal to $0$. However, I would like to find the expectation value of $S_T^2$ or even more the probability density of $S_T$. An advice how to find the associated Kolmogorov-Backward-Equation would also be fine.
Ignoring all mathematical rigor, this is easy enough to answer actually.
$S_T$ is a sum of (infinitely many) normal-distributed random variables. Thus it is itself normally distributed, obviously with mean equal zero.
The variance of $S_T$ can be computed as \begin{align} \langle S_T^2 \rangle &= \int_0^T \int_0^T t^2 s^2\langle X_t X_s\rangle\ dt\ ds \end{align}
Now if all the $X_t$ are (as you write) i.i.d. normally distributed with variance 1, then $\langle X_t X_s \rangle$ is 1 for $t=s$, and zero otherwise. This results in $\langle S_T^2\rangle=0$, which means $S_T=0$ (with probability one). I.e, your integral is almost always exactly zero.
Alternatively, you could increase the the variance of the individual $X_t$, for example to $\langle X_t X_s\rangle=\delta(t-s)$, then the answer would be more interesing, namely $\langle S_T^2 \rangle=T^5/5$.
Still another possibility is to take $X_t$ as as Wiener process, which means that each $X_t$ is still normal-distributed, but not independent. In that case, $S_T$ is still normal-distributed. Computing the variance is left to the reader :).