We present a systematic study of the relationship between Type Ia Supernova (SN Ia) properties, and the characteristics of their host galaxies, using a sample of 581 SNe Ia from the full Sloan Digital Sky Survey II (SDSS-II) SN Survey. We also investigate the effects of this on the cosmological constraints derived from SNe~Ia. Compared to previous studies, our sample is larger by a factor of $>4$, and covers a substantially larger redshift range (up to z~0.5), which is directly applicable to the volume of cosmological interest. We measure a significant correlation (>5\sigma) between the host-galaxy stellar-mass and the SN~Ia Hubble Residuals (HR). We find a weak correlation (1.4\sigma) between the host-galaxy metallicity as measured from emission lines in the spectra, and the SN~Ia HR. We also find evidence that the slope of the correlation between host-galaxy mass and HR is -0.11 $\mathrm{mag}/\log (M\mathrm{host}/M\odot )$ steeper in lower metallicity galaxies. We test the effects on a cosmological analysis using both the derived best-fitting correlations between host parameters and HR, and by allowing an additional free parameter in the fit to account for host properties which we then marginalize over when determining cosmological parameters. We see a shift towards more negative values of the equation of state parameter $w$, along with a shift to lower values of $\Omega m$ after applying mass or metallicity corrections. The shift in cosmological parameters with host-galaxy stellar-mass correction is consistent with previous studies. We find a best-fitting cosmology of $\Omega m=0.266-0.016+0.016$, $\Omega \Lambda =0.740-0.018+0.018$ and $w=-1.151-0.121+0.123$ (statistical errors only).