We investigate the stochastic nonlinear Schrödinger problem, characterized by a power-typenonlinearity and additive or multiplicative Gaussian noise, over a finite time horizon and a bounded one-dimensional domain. Based on the Galerkin method, truncation techniques and stopping times,we gain the unique existence of the variational solution and a priori estimates of the finite-dimensional equations. Thereafter, the existence and uniqueness of the variational solution of the infinite-dimensional problem are shown by using embedding and convergence theorems. In the case of linear multiplicative noise, we establish an equivalent pathwise nonlinear Schrödinger problem. Its smoothness results are transferred to the stochastic case and lead to properties we require to treat a corresponding problem of optimal control. Referring to the complex conjugated adjoint Schrödinger problem, we calculate a gradient formula of a given objective functional and a necessary optimality condition.