We present N/O abundance ratios, ionization parameters qion, and oxygen abundances O/H of eleven z~2 galaxies determined by the direct temperature Te method with [OIII]4363 and OIII]1665 lines, and investigate galaxy evolution from z~0 to 2 in conjunction with Te-method measurements of 208,529 SDSS galaxies and 9 green pea galaxies (GPs). One of these eleven z~2 galaxies is COSMOS-12805, which is a Lyα emitter at z=2.16. We have carried out three sets of spectroscopic observations for COSMOS-12805 with Subaru/FMOS, Keck/LRIS, and MOSFIRE. The rest-frame UV spectrum of COSMOS-12805 exhibits a reliable detection of OIII]1665 with a 6.8σ significance level. We identify that three out of our eleven z~2 galaxies clearly fall beyond the local average of N/O-O/H relations, while the rest of the z~2 galaxies have N/O ratios comparable with z~0 galaxies. The eleven z~2 galaxies place the upper limit of N/O ratio log(N/O)< -1.28 on average, suggesting that the N/O ratio evolves, if any, by <0.15 dex. We find that two of our z~2 galaxies with the significant BPT offsets show both N/O ratio and qion about 0.4-dex higher than the local average, while there exist z~0 green-pea galaxies whose N/O ratio and qion are comparable with these z~2 galaxies. Because a large fraction of our z~2 galaxies show qion and/or N/O-ratio excess(es) unlike the average local galaxies, we infer that galaxies with the BPT offsets emerge at z~2 and mostly disappear in the present-day universe, except for galaxies such as GPs.