Define $X(t) = \int_0^t \sigma(s) dW(s)$ where $W$ is a Wiener process aand $\sigma(s)$ some deterministic process. Informally this is the same as $dX(s) = \sigma(s) dW(s)$.
I wish to show that $X(t)$ is normally distributed with $0$ mean and variance $\int_0^t \sigma^2(s)ds$.
By letting $Z(t) = e^{u X(t)}$, we obtain from Ito's formula that $$dZ(t) = \frac{1}{2} \sigma^2(t) u^2 Z(t)dt + \sigma(t)uZ(t) dW(t). $$ In integral form, this is $$Z(t) = 1 + \int_0^t \frac{1}{2} \sigma^2(s) u^2 Z(s) ds + \int_0^t \sigma(s) uZ(s) dW(s). $$ Then we take expectations and after that, we differentiate in order to get a solvable DE.
First problem: Can I just move an expectation inside the first integral above?
Second problem: Is the expectation of the second integral 0? I know this to be true if the integrand is an adapted process which satisfies a particular integrability condition, but I am unsure if all that is satisfied here?
Third problem: Assuming the answers to above are yes, yes, we then want to take the derivative on both sides. On the right, the constant disappears but the integral remains. How do I take its derivative? Informally, I would presume it equals $\frac{1}{2} \sigma^2(t) u^2Z(t)$. Is this true?
Fourth problem: If the answer to above is yes, how do I solve the DE? I am unsure what to do with $\sigma$ when it depends on $t$.
It is correct that we obtain $$Z(t) = 1 + \int_0^t \frac{1}{2}\sigma(s)^2u^2 Z(s) ds + \int_0^t \sigma(s)u Z(s) dW(s), $$ $$ \Longrightarrow E(Z(t)) \equiv g(t) = 1 + \int_0^t \frac{1}{2}\sigma(s)^2 u^2 g(t) + \int_0^t \sigma(s)ug(t)dW(s),$$ $$\Longrightarrow g'(t) = \sigma(t)^2 \frac{1}{2}u^2 g(t) \Longrightarrow g(t) = e^{\int_0^t \frac{u^2}{2}\sigma(s)^2 ds}$$
Therefore, $E( e^{u X(t)}) = e^{\int_0^t \frac{u^2}{2}\sigma(s)^2 ds}.$ Now recall that a normally distributed variable $N$ with mean 0 and variance $\sigma^2$ has $E( e^{u N(t)}) = e^{ \frac{1}{2}u^2 \sigma^2 }.$ By comparison, we see that $X(t)$ is normally distributed with mean 0 and variance $\int_0^t \sigma(s)^2 ds$.
We used in the above calculations integrability and that the Ito-integral disappears once we take expectations. See your problem formulation in order to determine whether these assumptions are satisfied. They depend on the properties of $\sigma(s)$.