Chemodynamical evolution of the Milky Way disk - I. The solar vicinity

[Abridged] In this first paper of this series, we present a new approach for studying the chemo-dynamical evolution in disk galaxies, which consists of fusing disk chemical evolution models with compatible numerical simulations of galactic disks. This method avoids known star formation and chemical enrichment problems encountered in simulations. Here we focus on the Milky Way, by using a detailed thin-disk chemical evolution model and a simulation in the cosmological context, with dynamical properties close to those of our Galaxy. We show that, due to radial migration from mergers at high redshift and the central bar at later times, a sizable fraction of old metal-poor high-[alpha/Fe] stars reaches the solar vicinity. This naturally accounts for a number of recent observations related to both the thin and thick disks, despite the fact that we use thin-disk chemistry only. Although significant radial mixing is present, the slope in the age-metallicity relation is only weakly affected, with a scatter compatible with recent observational work. While we find a smooth density distribution in the [O/Fe]-[Fe/H] plane, we can recover the observed discontinuity by selecting particles according to kinematic criteria used in high-resolution samples to define the thin and thick disks. We outline a new method for estimating the birth place of the Sun and predict that the most likely radius lies in the range 4.4 < r < 7.7 kpc (for a location at r = 8 kpc). A new, unifying model for the Milky Way thick disk is offered, where both mergers and radial migration play a role at different stages of the disk evolution. We show that in the absence of early-on massive mergers the vertical velocity dispersion of the oldest stars is underestimated by a factor of ~2 compared with observations. We can, therefore, argue that the Milky Way thick disk is unlikely to have been formed through a quiescent disk evolution.