We present the discovery and spectrophotometric characterization of a large sample of 164 faint ($iAB$ $\sim$ $23$-$25$ mag) star-forming dwarf galaxies (SFDGs) at redshift $0.13$ $\le z\le$ $0.88$ selected by the presence of bright optical emission lines in the VIMOS Ultra Deep Survey (VUDS). We investigate their integrated physical properties and ionization conditions, which are used to discuss the low-mass end of the mass-metallicity relation (MZR) and other key scaling relations. We use optical VUDS spectra in the COSMOS, VVDS-02h, and ECDF-S fields, as well as deep multiwavelength photometry, to derive stellar masses, star formation rates (SFR) and gas-phase metallicities. The VUDS SFDGs are compact (median $re$ $\sim$ $1.2$ kpc), low-mass ($M\ast$ $\sim$ $107-109$ $M\odot$) galaxies with a wide range of star formation rates (SFR($H\alpha$) $\sim 10-3-101$ $M\odot /yr$) and morphologies. Overall, they show a broad range of subsolar metallicities (12+log(O/H)=$7.26$-$8.7$; $0.04$ $\lesssim Z/Z\odot \lesssim$ $1$). The MZR of SFDGs shows a flatter slope compared to previous studies of galaxies in the same mass range and redshift. We find the scatter of the MZR partly explained in the low mass range by varying specific SFRs and gas fractions amongst the galaxies in our sample. Compared with simple chemical evolution models we find that most SFDGs do not follow the predictions of a "closed-box" model, but those from a gas regulating model in which gas flows are considered. While strong stellar feedback may produce large-scale outflows favoring the cessation of vigorous star formation and promoting the removal of metals, younger and more metal-poor dwarfs may have recently accreted large amounts of fresh, very metal-poor gas, that is used to fuel current star formation.