We present new low-resolution (R ~ 800) optical spectra of 22 Galactic planetary nebulae (PNe) with Spitzer spectra. These data are combined with recent optical spectroscopic data available in the literature to construct representative samples of compact (and presumably young) Galactic disc and bulge PNe with Spitzer spectra. Attending to the nature of the dust features - C-rich, O-rich, and both C- and O-rich dust features (or double chemistry) - seen in their Spitzer spectra, the Galactic disc and bulge PNe are classified according to four major dust types (oxygen chemistry or OC, carbon chemistry or CC, double chemistry or DC, featureless or F) and subtypes (amorphous and crystalline, and aliphatic and aromatic), and their Galactic distributions are presented. Nebular gas abundances of He, N, O, Ne, S, Cl, and Ar, as well as plasma parameters (e.g. Ne, Te) are homogeneously derived by using the classical empirical method. We study the median chemical abundances and nebular properties in Galactic disc and bulge PNe depending on their Spitzer dust types and subtypes. The differences and similarities between PNe in the Galactic disc and bulge are reported. In particular, the median abundances for the major Spitzer dust types CC and OC are representative of the dominant dust subtype (which are different in both Galactic environments), while these values in DC PNe are representative of the two DC subtypes. A comparison of the derived median abundance patterns with AGB nucleosynthesis predictions mainly show that i) DC PNe, both with amorphous and crystalline silicates, display high-metallicity (solar/supra-solar) and the highest He abundances and N/O abundance ratios, suggesting relatively massive (~3-5 M⊙) hot bottom burning AGB stars as progenitors; ii) PNe with O-rich and C-rich unevolved dust (amorphous and aliphatic) seem to evolve from subsolar metallicity (z ~ 0.008) and lower mass (<3 M⊙) AGB stars; iii) a few O-rich PNe and a significant fraction of C-rich PNe with more evolved dust (crystalline and aromatic, respectively) display chemical abundances similar to DC PNe, suggesting that they are related objects. A comparison of the derived nebular properties with predictions from models combining the theoretical central star evolution with a simple nebular model is also presented. Finally, a possible link between the Spitzer dust properties, chemical abundances, and evolutionary status is discussed. The appendices are available in electronic form at http://www.aanda.orgTables 2 and 3 are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/567/A12