Wave statistical properties and occurrence of extreme and rogue waves in crossing sea states are investigated. Compared to previous studies a more extensive set of crossing sea states are investigated, both with respect to spectral shape of the individual wave systems and with respect to the crossing angle and separation in peak frequency of the two wave systems. It is shown that, because of the effects described by Piterbarg, for a linear sea state the expected maximum crest elevation over a given surface area depends on the crossing angle so that the expected maximum crest elevation is largest when two wave systems propagate with a crossing angle close to 90°. It is further shown by nonlinear phase-resolving numerical simulations that nonlinear effects have an opposite effect, such that maximum sea surface kurtosis is expected for relatively large and small crossing angles, with a minimum around 90°, and that the expected maximum crest height is almost independent of the crossing angle. The numerical results are accompanied by analysis of the modulational instability of two crossing Stokes waves, which is studied using the Zakharov equation so that, different from previous studies, results are valid for arbitrary-bandwidth perturbations. It is shown that there is a positive correlation between the value of kurtosis in the numerical simulations and the maximum unstable growth rate of two crossing Stokes waves, even for realistic broadband crossing sea states.