We present near-IR spectroscopy in the J- and H-bands for a large sample of 243 X-ray-selected, moderate-luminosity Type 1 active galactic nuclei (AGNs) in the COSMOS, SXDS, and E-CDF-S survey fields using the multi-object spectrograph Subaru/FMOS. Our sample covers the redshift range 0.5<=z<=3.0 and X-ray luminosity range of 10^43^<=L_[2-10keV]_<=10^45^erg/s. We provide emission-line properties and derived virial black hole mass estimates, bolometric luminosities, and Eddington ratios, based on H{alpha} (211), H{beta} (63), and MgII (4). We compare line widths, luminosities, and black hole mass estimates from H{alpha} and H{beta}, and augment these with commensurate measurements of MgII and CIV detected in optical spectra. We demonstrate the robustness of using H{alpha}, H{beta}, and MgII as reliable black hole mass estimators for high-z moderate-luminosity AGNs, while the use of CIV is prone to large uncertainties (>=0.4dex). We extend a recently proposed correction based on the CIV blueshift to lower luminosities and black hole masses. While our sample shows an improvement in their CIV black hole mass estimates, the deficit of high blueshift sources reduces its overall importance for moderate-luminosity AGNs compared to the most luminous quasars. In addition, we revisit luminosity correlations between Lbol, L_[2-10keV]_, L[OIII], L5100, and LH{alpha} and find them to be consistent with a simple empirical model, based on a small number of well-established scaling relations. Finally, we highlight our highest redshift AGN, CID 781, at z=4.6, which has the lowest black hole mass (~10^8^M_{sun}_) among current near-IR samples at this redshift and is in a state of fast growth.