Given a random zero-mean gaussian random variable $X(t)$ with parameter $t$, such that $E [X(t) X(t^\prime)] = \sigma^2 (t) \delta_{tt^\prime}$, is it possible to produce a single gaussian random variable that is the weighted sum of all $X(t)$ by a non-random function $f(t)$? In other words, is $Z=\int_0^T f(t) X(t) dt$ well defined, and if so, is $Z$ a gaussian normal random variable, and what is its variance?
My naive attempt is to discretize the integral \begin{equation} Z = \int_0^T f(t) X(t) dt \approx \sum_{n=0}^{N-1} f(n \delta T) X(n \delta T) \delta T \end{equation} where $\delta T = T/N$. Now, if we take the simplest limit where $\sigma(t) = \sigma_0$ and $f(t) = 1$, this reduces to \begin{eqnarray} Z & = & \frac{T}{N}\sum X_n \end{eqnarray} But this is a gaussian random variable with $\sigma = \sqrt{N} \delta T \sigma_0 = T \sigma_0 / \sqrt{N}$, which has $\sigma \rightarrow 0$ as $N \rightarrow \infty$.
If I were to take a guess, I'd assume $Z$ would be gaussian and have variance $\sigma^2 = \int_0^T f^2(t) \sigma^2(t) dt$, but I have been unable to show this.
Kudos for your logic in studying this problem (except that your $\sigma$ goes to $0$ when $N\to\infty$). Indeed, as you realized, the construction of the object $Z$ runs into serious problems, which are of two kinds mainly.
First, an i.i.d. process $(X(t))_{t\in\mathbb R_+}$ is a wild beast, for example, which probability space are we going to use? Second, the way to define the integral $\int\limits_0^TX(t)\mathrm dt$ is not obvious at all. The Riemann way is doomed since every upper Darboux sum is $+\infty$ and every lower Darboux sum is $-\infty$, and the Lebesgue way runs into measurability issues.
Note that stochastic integrals are based on quite different processes, for instance on a Brownian motion $(W_t)_{t\in\mathbb R_+}$, which is far from being i.i.d. To begin with, its paths being almost surely continuous, this process is entirely determined by $(W_t)_{t\in\mathbb Q_+}$, that is, a countable collection of random variables--and now one can begin to work...