We prove existence and uniqueness of solutions to a class of stochastic semilinear evolution equations with a monotone nonlinear drift term and multiplicative noise, considerably extending corresponding results obtained in previous work of ours. In particular, we assume the initial datum to be only measurable and we allow the diffusion coefficient to be locally Lipschitz-continuous. Moreover, we show, in a quantitative fashion, how the finiteness of the $p$-th moment of solutions depends on the integrability of the initial datum, in the whole range $p \in ]0,\infty[$. Lipschitz continuity of the solution map in $p$-th moment is established, under a Lipschitz continuity assumption on the diffusion coefficient, in the even larger range $p \in [0,\infty[$. A key role is played by an Itô formula for the square of the norm in the variational setting for processes satisfying minimal integrability conditions, which yields pathwise continuity of solutions. Moreover, we show how the regularity of the initial datum and of the diffusion coefficient improves the regularity of the solution and, if applicable, of the invariant measures.