  We search for local extremely metal-poor galaxies (EMPGs), selecting
photometric candidates by broadband color excess and machine-learning
techniques with the SDSS photometric data. After removing stellar contaminants
by shallow spectroscopy with Seimei and Nayuta telescopes, we confirm that
three candidates are EMPGs with 0.05--0.1 $Z_\odot$ by deep Magellan/MagE
spectroscopy for faint {\sc[Oiii]}$\lambda$4363 lines. Using a statistical
sample consisting of 105 spectroscopically-confirmed EMPGs taken from our study
and the literature, we calculate cross-correlation function (CCF) of the EMPGs
and all SDSS galaxies to quantify environments of EMPGs. Comparing another CCF
of all SDSS galaxies and comparison SDSS galaxies in the same stellar mass
range ($10^{7.0}-10^{8.4} M_\odot$), we find no significant ($>1\sigma$)
difference between these two CCFs. We also compare mass-metallicity relations
(MZRs) of the EMPGs and those of galaxies at $z\sim$ 0--4 with a steady
chemical evolution model and find that the EMPG MZR is comparable with the
model prediction on average. These clustering and chemical properties of EMPGs
are explained by a scenario of stochastic metal-poor gas accretion on
metal-rich galaxies showing metal-poor star formation. Extending the broadband
color-excess technique to a high-$z$ EMPG search, we select 17 candidates of
$z\sim$ 4--5 EMPGs with the deep ($\simeq30$ mag) near-infrared JWST/NIRCam
images obtained by ERO and ERS programs. We find galaxy candidates with
negligible {\sc[Oiii]}$\lambda\lambda$4959,5007 emission weaker than the local
EMPGs and known high-$z$ galaxies, suggesting that some of these candidates may
fall in 0--0.01 $Z_\odot$, which potentially break the lowest metallicity limit
known to date.
