We present a homogeneous set of accurate atmospheric parameters for a complete sample of very and extremely metal-poor stars in the dwarf spheroidal galaxies (dSphs) Sculptor, Ursa Minor, Sextans, Fornax, Bootes I, Ursa Major II, and Leo IV. We also deliver a Milky Way (MW) comparison sample of giant stars covering the -4 < [Fe/H] < -1.7 metallicity range. We show that, in the [Fe/H] > -3.5 regime, the non-local thermodynamic equilibrium (NLTE) calculations with non-spectroscopic effective temperature (Teff) and surface gravity (log~g) based on the photometric methods and known distance provide consistent abundances of the Fe I and Fe II lines. This justifies the Fe I/Fe II ionisation equilibrium method to determine log g for the MW halo giants with unknown distance. The atmospheric parameters of the dSphs and MW stars were checked with independent methods. In the [Fe/H] > -3.5 regime, the Ti I/Ti II ionisation equilibrium is fulfilled in the NLTE calculations. In the log~g - Teff plane, all the stars sit on the giant branch of the evolutionary tracks corresponding to [Fe/H] = -2 to -4, in line with their metallicities. For some of the most metal-poor stars of our sample, we hardly achieve consistent NLTE abundances from the two ionisation stages for both iron and titanium. We suggest that this is a consequence of the uncertainty in the Teff-colour relation at those metallicities. The results of these work provide the base for a detailed abundance analysis presented in a companion paper.
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