  We report a systematic measurement of cumulants, $C_{n}$, for net-proton,
proton and antiproton, and correlation functions, $\kappa_n$, for proton and
antiproton multiplicity distributions up to the fourth order in Au+Au
collisions at $\sqrt{s_{\mathrm {NN}}}$ = 7.7, 11.5, 14.5, 19.6, 27, 39, 54.4,
62.4 and 200 GeV. The $C_{n}$ and $\kappa_n$ are presented as a function of
collision energy, centrality and kinematic acceptance in rapidity, $y$, and
transverse momentum, $p_{T}$. The data were taken during the first phase of the
Beam Energy Scan (BES) program (2010 -- 2017) at the Relativistic Heavy Ion
Collider (RHIC) facility. The measurements are carried out at midrapidity ($|y|
<$ 0.5) and transverse momentum 0.4 $<$ $p_{\rm T}$ $<$ 2.0 GeV/$c$, using the
STAR detector at RHIC. We observe a non-monotonic energy dependence
($\sqrt{s_{\mathrm {NN}}}$ = 7.7 -- 62.4 GeV) of the net-proton $C_{4}$/$C_{2}$
with the significance of 3.1$\sigma$ for the 0-5\% central Au+Au collisions.
This is consistent with the expectations of critical fluctuations in a
QCD-inspired model. Thermal and transport model calculations show a monotonic
variation with $\sqrt{s_{\mathrm {NN}}}$. For the multiparticle correlation
functions, we observe significant negative values for a two-particle
correlation function, $\kappa_2$, of protons and antiprotons, which are mainly
due to the effects of baryon number conservation. Furthermore, it is found that
the four-particle correlation function, $\kappa_4$, of protons plays a role in
determining the energy dependence of proton $C_4/C_1$ below 19.6 GeV, which
cannot be solely understood by the negative values of $\kappa_2$ for protons.
