Slight generalisation of the distribution of Brownian integral I think I have seen once that if the processes $\sigma$ and $W$, a Brownian motion, are independent then one has that 
$$
E \left[\exp \left(iu\int_t^T \sigma_s \, dW_s\right) \mid \mathcal{F} \right] = E\left[\exp \left(- \frac{1}{2} u^2 \int_t^T \sigma_s^2 \, ds \right) \mid \mathcal{F} \right]
$$
for $\mathcal{F}$ the $\sigma$-algebra generated by $(\sigma_s)_{ s \leq T}$. I am not sure the formulation is completely correct though, it is purely from memory. 
Is this true? - could one help with a reference?
If $\sigma$ is of bounded variation, is it then true that the integral can be defined without use of the the whole usual "limit in $L^2$ construction" of the stochastic integral. If so what is a reference for that?
Edit: I've found a brief discussion of the second fact see here
 A: First consider the case that $\sigma$ is a simple function, i.e.
$$\sigma(s) =\sum_{j=1}^n 1_{[t_{j-1},t_j)} \xi_j \tag{1}$$
where $(\xi_j)_{j=1,\ldots,n}$ are random variables independent from the Brownian motion $W$. Without loss of generality, we may assume that $t_k = t$ for some $k \in \{1,\ldots,n\}$ (otherwise we add the point to the partition). Then
$$\begin{align*} \mathbb{E} \left( \exp \left[ \imath \, u \int_t^T \sigma_s \, dW_s \right] \mid \mathcal{F} \right) &= \mathbb{E} \left( \exp \left[ \imath u \sum_{j=k+1}^n \xi_j (W_{t_j}-W_{t_{j-1}} \right] \mid \mathcal{F} \right). \end{align*}$$
Since $W$ is independent of $\mathcal{F}$ and $\xi_j$ measurable with respect to $\mathcal{F}$, we get
$$\begin{align*} \mathbb{E} \left( \exp \left[ \imath \, u \int_t^T \sigma_s \, dW_s \right] \mid \mathcal{F} \right) &= \mathbb{E} \left( \exp \left[ \imath \, u \sum_{j=k+1}^n y_j (W_{t_j}-W_{t_{j-1}}) \right] \right) \bigg|_{y_j = \xi_j} \\ &= \exp \left( - \frac{1}{2} u^2 \sum_{j=k+1}^n y_j^2 (t_j-t_{j-1}) \right) \bigg|_{y_j = \xi_j} \\ &= \exp \left( - \frac{1}{2} u^2 \int_t^T \sigma^2(s) \, ds \right). \end{align*}$$
For general $\sigma$, approximate $\sigma$ by suitable simple functions of the form $(1)$ and use the dominated convergence theorem to conclude that
$$ \mathbb{E} \left( \exp \left[ \imath \, u \int_t^T \sigma_s \, dW_s \right] \mid \mathcal{F} \right) = \exp \left( - \frac{1}{2} u^2 \int_t^T \sigma^2(s) \, ds \right).$$

Concerning your second question: For functions of bounded variation, a stochastic integral can be defined by integration by parts, i.e. by setting
$$\int_0^t \sigma(s) \,d W_s := \sigma(t) W(t) - \int_0^t W(s) \, d\sigma(s).$$
The integral at the right-hand side is well-define since $\sigma$ is of bounded variation. This leads to the so-called Paley-Wiener-Zygmund integral (which are a special case of the Itô integral); see e.g.
Rene Schilling, Lothar Partzsch: Brownian Motion - An Introduction to Stochastic Processes, Remark 13.5
for a (short) introduction.
