We perform population synthesis studies of different types of neutron stars (thermally emitting isolated neutron stars, normal radio pulsars, magnetars) taking into account the magnetic field decay and using results from the most recent advances in neutron star cooling theory. For the first time, we confront our results with observations using simultaneously the Log N - Log S distribution for nearby isolated neutron stars, the Log N - Log L distribution for magnetars, and the distribution of radio pulsars in the P - Ṗ diagram. For this purpose, we fix a baseline neutron star model (all microphysics input), and other relevant parameters to standard values (velocity distribution, mass spectrum, etc.), only allowing to vary the initial magnetic field strength. We find that our theoretical model is consistent with all sets of data if the initial magnetic field distribution function follows a log-normal law with <log(B0/[G])>∼13.25 and σ logB0∼0.6. The typical scenario includes about 10% of neutron stars born as magnetars, significant magnetic field decay during the first million years of a NS life (only about a factor of 2 for low field neutron stars but more than an order of magnitude for magnetars), and a mass distribution function dominated by low mass objects. This model explains satisfactorily all known populations. Evolutionary links between different subclasses may exist, although robust conclusions are not yet possible. We apply the obtained field distribution and the model of decay to study long-term evolution of neuton stars till the stage of accretion from the interstellar medium. It is shown that though the subsonic propeller stage can be relatively long, initially highly magnetized neutron stars (B0>∼10 13G) reach the accretion regime within the Galactic lifetime if their kick velocities are not too large. The fact that in previous studies made >10 years ago, such objects were not considered results in a slight increase of the Accretor fraction in comparison with earlier conclusions. Most of the neutron stars similar to the Magnificent seven are expected to become accreting from the interstellar medium after few billion years of their evolution. They are the main predecestors of accreting isolated neutron stars.