  Measurements of exclusive $J/\psi$, $\psi(2s)$, and electron-positron
($e^{+}e^{-}$) pair photoproduction in Au$+$Au ultra-peripheral collisions are
reported by the STAR experiment at $\sqrt{s_{_\mathrm{NN}}}=200~\rm{GeV}$. We
report several first measurements at the Relativistic Heavy-Ion Collider, which
are i) $J/\psi$ photoproduction with large momentum transfer up to
$2.2~\rm{(GeV/c)^{2}}$, ii) coherent $J/\psi$ photoproduction associated with
neutron emissions from nuclear breakup, iii) the rapidity dependence of
incoherent $J/\psi$ photoproduction, iv) the $\psi(2s)$ photoproduction cross
section at mid-rapidity, and v) $e^{+}e^{-}$ pair photoproduction up to high
invariant mass of 6 $\rm{GeV/c^2}$. For measurement ii), the coherent $J/\psi$
total cross section of $\gamma^{\ast} + \rm{Au} \rightarrow J/\psi + \rm{Au}$
as a function of the center-of-mass energy $W_{\rm{\gamma* N}}$ has been
obtained without photon energy ambiguities. The data are quantitatively
compared with the Monte Carlo models STARlight, Sartre, BeAGLE, and theoretical
calculations of gluon saturation with color glass condensate, nuclear shadowing
with leading twist approximation, Quantum Electrodynamics, and the
Next-to-Leading Order perturbative QCD. At the photon-nucleon center-of-mass
energy of 25.0 GeV, the coherent and incoherent $J/\psi$ cross sections of Au
nuclei are found to be $71\pm10\%$ and $36\pm7\%$, respectively, of that of
free protons. These data provide an important experimental constraint for
nuclear parton distribution functions and a unique opportunity to advance the
understanding of the nuclear modification effect at the top RHIC energy.
