In this paper, the author studies the stochastic dispersive equation with linear multiplicative noise \[ \begin{cases} dX(t)=i P(x,D) X(t) dt+F(t) dt - \mu X(t) dt +X(t) d W(t)\,, \quad t \in (0,T)\,, \\ X(0)=X_0.\tag{1} \end{cases} \] Here, \(X\) and \(F\) are complex-valued functions on \([0,T] \times \mathbb{R}^d, T \in (0,\infty)\), \(P(x,D)\) is a pseudodifferential operator of order \(m \geq 2\) and \(D=-i (\delta_{x_1},\ldots,\delta_{x_n})\). Furthermore, \(W\) is a coloured Wiener process taken to be of the form \[ W(t,x)=\sum_{j=1}^{N} \mu_j e_j(x) \beta_j(t), \] for \(e_j\) real-valued functions and \(\beta_j\) independent real Brownian motions on a probability space \((\Omega,\mathcal{F}, \mathbb{P})\) with natural filtration \((\mathcal{F}_t)_{t \geq 0}\). Moreover, \[ \mu(x)=\frac{1}{2} \sum_{j=1}^{N} |\mu_j|^2 e_j^2(x). \] The main result of the paper is to prove pathwise Strichartz and local smoothing estimates for (1), under suitable assumptions. This result is then applied to prove well-posedness for a general class of stochastic nonlinear Schrödinger equations with variable coefficients and lower order perturbations. Furthermore, one obtains \(\mathbb{P}\)-integrability of such stochastic nonlinear Schrödinger equations with constant coefficients in both the mass- and energy-subcritical regime.