We present the fully parallel chemo-dynamical Tree/SPH code GEAR, which allows us to perform high resolution simulations with detailed chemical diagnostics. Starting from the public version of Gadget-2, we included the complex treatment of the baryon physics: gas cooling, star formation law, chemical evolution, and supernova feedback. We qualified the performances of GEAR in the case of dwarf spheroidal galaxies (dSphs) galaxies. Our code GEAR conserves the total energy budget of the systems to better than 5% over 14 Gyr and provides an excellent convergence of the results with numerical resolution. We showed that models of dSphs in a static Euclidean space, where the expansion of the universe is neglected are valid. In addition, we tackled some existing open questions in the field, such as the stellar mass fraction of dSphs and its link to the predicted dark matter halo mass function, the effect of supernova feedback, the spatial distribution of the stellar populations, and the origin of the diversity in star formation histories and chemical abundance patterns. Strong supernova-driven winds seem incompatible with the observed metallicities and luminosities. Despite newly formed stars being preferentially found in the galaxy central parts, turbulent motions in the gas can quickly erase any metallicity gradient. The diversity in properties of dSph are related to a range of total masses, as well as a range of dispersion in the central densities, which is also seen in the halos emerging from a Lambda CDM cosmogony.