We study the statistical and phase properties of the output states of a Mach-Zehnder interferometer with a nonlinear Kerr medium in one of its arms. The combination of nonlinearity and displacement produced by the interferometer generates output states (displaced Kerr states) that, unlike those derived from a Kerr medium alone (Kerr states), do not necessarily retain the photon statistics of the initially coherent field. As the parameter characterizing the nonlinear medium is varied, the statistics of the output field vary from sub-Poissonian to super-Poissonian. The photon-number distributions of these states are rationalized by examining their quasiprobability distributions, which vary from banana-shaped for sub-Poissonian statistics to complicated interference structures for highly super-Poissonian statistics. The number-phase properties of the Kerr and displaced Kerr states are studied for the case of small photon numbers (relevant to cavity quantum electrodynamics) using a methodology based on the Pegg-Barnett formalism. In particular, we determine the range of parameters for which the states are minimum-uncertainty states and whether they are squeezed with respect to photon number and phase, drawing a distinction between a sub-Poissonian nature and number squeezing. Finally, we find that sub-Poissonian statistics can coexist with quadrature squeezing for a range of parameters.