Dust properties and star formation of approximately a thousand local galaxies.

[ABRIDGED] We derive the dust properties for 753 local galaxies and examine how these relate to some of their physical properties. We model their global dust-SEDs, treated statistically as an ensemble within a hierarchical Bayesian dust-SED modeling approach. The model-derived properties are the dust masses (Mdust), the average interstellar radiation field intensities (Uav), the mass fraction of very small dust grains ('QPAH' fraction), as well as their standard deviations. In addition, we use mid-IR observations to derive SFR and Mstar, quantities independent of the modeling. We derive distribution functions of the properties for the galaxy ensemble and per galaxy type. The mean value of Mdust for the ETGs is lower than that for the LTGs and IRs, despite ETGs and LTGs having Mstar spanning across the whole range observed. The Uav and 'QPAH' fraction show no difference among different galaxy types. When fixing Uav to the Galactic value, the derived 'QPAH' fraction varies across the Galactic value (0.071). The sSFR increases with galaxy type, while this is not the case for the dust-sSFR (=SFR/Mdust), showing an almost constant SFE per galaxy type. The galaxy sample is characterised by a tight relation between Mdust and Mstar for the LTGs and Irs, while ETGs scatter around this relation and tend towards smaller Mdust. While the relation indicates that Mdust may fundamentally be linked to Mstar, metallicity and Uav are the second parameter driving the scatter, which we investigate in a forthcoming work. We use the extended KS law to estimate Mgas and the GDR. The Mgas derived from the extended KS law is on average ~20% higher than that derived from the KS law, and a large standard deviation indicates the importance of the average SF present to regulate star formation and gas supply. The average GDR for the LTGs and IRs is 370, while including the ETGs gives an average of 550. [ABRIDGED]