I'm looking for a proof/calculation for the following result I've seen quoted a few times in papers and notes but never explained.
Let $\mu$ denote the Weiner measure on the space of real valued continuous paths $C[0,1]$. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous deterministic function and consider the Ito integral
$$X_t = \int_0^t f(s)\mathrm{d}W_s$$
We can say that $X_t$ is a functional on the space of paths, i.e. $X_t = F(\omega)$, because you give it a (random) continuous function and it will return a number. I'm hesitant to interpet the Ito integral pathwise but I have seen things written such as $X_t(\omega) = \int_0^t f(s)\mathrm{d}\omega_s$ whatever this means, regardless one can consider expressions such as:
$$\mathbb{E}_\mu[X_t] = \int_{C[0,1]}X_t(\omega)\mu( \mathrm{d} \omega)$$
I believe that it so happens that $\mathbb{E}_\mu[X_t] = 0$ analogous (possibly equivalent) to the fact that the Ito integral is a martingale with expectation $0$. I can't recall but pretty sure I've also seen something like $\mathbb{E}_\mu[X_t^2] = \int_0^t f(s)^2ds$, the Ito isometry.
Is this correct? And if so how can I show this? The only explicit calculations I've ever seen for integrals involving $\mu$ is when we have the case that $F(\omega) = g(\omega(t_1), \ldots, \omega(t_n))$ for some times $t_i$ and $g : \mathbb{R}^n \rightarrow \mathbb{R}$, all other functions have to be treated as an abstract limit of approximations of the type $g$.
We know from countless textbooks introducing Ito calculus that for continuous and deterministic, hence bounded, $f$ on $[0,t]\,:$ $$ X_t=\lim_{||\Pi||\to0}\sum_{j=0}^{n-1}f(t_j)(W_{t_{j+1}}-W_{t_j})\,\quad\text{ in }L^2(\Omega,\cal A,\mathbb P)\,. $$ Therefore, there is a subsequence that converges to $X_t$ $\quad\mathbb P$-a.s.
The rest of OP is correct. Use the martingale property and the well-known Ito isometry.
Book: Karatzas & Shreve, Brownian Motion and Stochastic Calculus or many others.