We present near-IR spectroscopy in J- and H-band for a large sample of 243 X-ray selected moderate-luminosity type-1 AGN 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 an X-ray luminosity range of $10^{43}lesssim L_X lesssim 10^{45}$~erg s$^{-1}$. 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 AGN, while the use of CIV is prone to large uncertainties (>0.4 dex). 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 AGN, compared to the most luminous quasars. In addition, we revisit luminosity correlations between $L_{rm{bol}}$, $L_X$, $L_{rm{[OIII]}}$, $L_{5100}$ and $L_{rm{H}alpha}$ and find them to be consistent with a simple empirical model, based on a small number of well-established scaling relations. We finally highlight our highest redshift AGN, CID 781, at z=4.6 which shows the lowest black hole mass ($sim10^8$M$_odot$) among current near-IR samples at this redshift, and is in a state of fast growth.
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