We present the properties of 17 faint Lyman-α emitting galaxies (LAEs) at z > 5.8 from the JWST Advanced Deep Extragalactic Survey (JADES) spectroscopic data in the Hubble Ultra Deep Field/GOODS-S. These LAEs span a redshift range z ≈ 5.8 − 8.0 and a UV magnitude range M_UV ≈ −17 to −20.6, with the Lyα equivalent width (EW) in the range ≈25 − 350 Å. The detection of other rest-optical emission lines in the spectra of these LAEs enables the determination of accurate systemic redshifts and Lyα velocity offsets, as well as the physical and chemical composition of their stars and interstellar media. These faint LAEs are consistent with metal-poor systems with high ionization parameters, similar to the general galaxy population at z > 6. We measured an average ionizing photon production efficiency, log(ξ_ion/erg⁻¹ Hz) ≈ 25.57 across our LAEs, which does not evolve strongly with redshift. We report an anti-correlation between the Lyα escape fraction and the velocity offset from systemic redshift, consistent with model expectations. We further find that the strength and velocity offset of Lyα are neither correlated with galaxy spectroscopic properties nor with ξ_ion. We find a decrease in Lyα escape fractions with redshift, indicative of decreasing sizes of ionized bubbles around LAEs at high redshifts. We used a range of galaxy properties to predict Lyman continuum escape fractions for our LAEs, finding that the ionizing photon output into the intergalactic medium from our LAEs remains roughly constant across the observed Lyα EW, showing a mild increase at fainter UV magnitudes and at higher redshifts. We derived correlations between the ionizing photon output from LAEs and their UV magnitudes, Lyα strengths and redshifts, which can be used to constrain the ionizing photon contribution of LAEs at z > 6 towards cosmic reionization.