Complete, accurate, and precise catalogues of exoplanet host star (EHS) properties are essential to derive high-quality exoplanet parameters, and, thus, to study individual planets, planet populations, planet formation, and their Galactic context. This article aims at homogeneously parameterising EHS and their exoplanets, selected from the Encyclopedia of Exoplanetary Systems and the NASA Exoplanets Archive, thanks to Gaia astrometric, photometric and GSP-Spec spectroscopic data, complemented by some ground-based spectroscopic survey information. From the atmospheric parameters of 2573 EHS, we computed their luminosity, radius, and mass, with no prior assumption from stellar evolution models. Their Galactic positions, kinematic and orbital properties are also derived. We then re-scale the mass and radius of 3556 exoplanets, fully consistently with the stellar data (when available). The Gaia spectroscopic stellar effective temperatures, luminosities and radii rather well agree with literature values but are more precise. In particular, stellar radii are derived with typically less than 3% uncertainty (instead of ~8% in the literature); this reduces significantly the uncertainty on the planetary radii and allows a finer analysis of the decrease of planet number around 1.8R_{Earth}_ (evaporation valley). Larger differences are however found for the masses that are more difficult to estimate by any methods. The EHS population is rather diverse in chemical and Galactic properties, although they are all found in the Solar vicinity, close to the Local spiral arm. Most EHS belong to the thin disc but some older thick disc and halo members are also identified. For the less massive planets (log(Mp/MJup)<~-0.6), the average planet radius increases with the metallicity of the host star. For giant planets, a dichotomy between dense and inflated planets is found. Denser planets (Rp<~1.1R_Jup_) tend to be more massive as the metallicity of the host star increases, while inflated planets are more massive for less metallic hosts. If confirmed, this bimodality implies that the diversity of giant exoplanets depends on their Galactic birth locus, with dense giant planets being more numerous than inflated ones when [M/H] is higher than ~1.5 times Solar, as in the central Milky Way regions. The Gaia spectroscopic catalogue of exoplanets and their host stars is large, homogeneous, and precise. It thus should be a valuable added-value for planetary studies. Since it is based on literature data, it could also easily be updated thanks to future Gaia data releases and other space and ground-based surveys.