We present JWST NIRSpec prism spectroscopy of gravitationally lensed galaxies at zrsim9 found behind the massive galaxy cluster Abell 2744 in the UNCOVER Cycle 1 Treasury Program. We confirm the source redshift via emission lines and/or the Lyα break feature for ten galaxies at z=8.50-13.08 down to M_UV=-17.3. We achieve a high confirmation rate of 100% for z>9 candidates reported in Atek et al. (2023). Using six sources with multiple emission line detections, we find that the offset of the redshift estimates between the lines and the Lyα break alone with prism can be as large as ±0.2, raising caution in designing future follow-up spectroscopy for the break-only sources. With spec-z confirmed sources in UNCOVER and the literature, we derive lower limits on the rest-frame ultraviolet (UV) luminosity function (LF) at z≃9-12 and find these lower limits to be consistent with recent photometric measurements. We identify at least two unambiguous and several possible active galactic nucleus (AGN) systems based on X-ray emission, broad line (BL) Hβ, high ionization line (e.g., NIV]1487, CIV1549) detections, and excess in UVLF. This requires the AGN LFs at z≃ 9-10 to be comparable or even higher than the X-ray AGN LF estimated at z∼6 and indicates a plausible cause of the high abundance of z>9 galaxies claimed in recent photometric studies may be AGNs. One UV-luminous source is confirmed at the same redshift as a dusty BL AGN at z=8.50 with a physical separation of 380 kpc in the source plane. These two sources show blueward Lyα line or continuum emission, suggesting that they reside in the same ionized bubble with a radius of 7.69±0.18 pMpc. Our results imply that AGNs have a non-negligible contribution to cosmic reionization.