  Proton number fluctuation is sensitive observable to search for the QCD
critical point in heavy-ion collisions. In this paper, we studied rapidity
acceptance dependence of the proton cumulants and correlation functions in most
central Au+Au collisions at $\sqrt{s_\mathrm{NN}} = 5$ GeV from a microscopic
hadronic transport model (JAM). At mid-rapidity, we found the effects of
resonance weak decays and hadronic re-scattering on the proton cumulants and
correlation functions are small, but those effects get larger when further
increasing the rapidity acceptance. On the other hand, we found the baryon
number conservation is a dominant background effect on the rapidity acceptance
dependence of proton number fluctuations. It leads to a strong suppression of
cumulants and cumulant ratios, as well as the negative proton correlation
functions. We also studied those two effects on the energy dependence of
cumulant ratios of net-proton distributions in most central Au+Au collisions at
$\sqrt{s_\mathrm{NN}} = 5-200$ GeV from JAM model. This work can serve as a
non-critical baseline for future QCD critical point search in heavy-ion
collisions at high baryon density region.
