  The gravitational-wave (GW) events, produced by the coalescence of binary
neutron-stars (BNS), can be treated as the standard sirens to probe the
expansion history of the Universe, if their redshifts could be determined from
the electromagnetic observations. For the high-redshift ($z\gtrsim 0.1$)
events, the short $\gamma$-ray bursts (sGRBs) and the afterglows are always
considered as the primary electromagnetic counterparts. In this paper, by
investigating various models of sGRBs and afterglows, we discuss the rates and
distributions of BNS mergers' multi-messenger observations with GW detectors in
second-generation (2G), 2.5G, 3G era with the detectable sGRBs and the
afterglows. For instance, for Cosmic Explorer GW detector, the rate is about
(300-3500) per year with GECAM-like detector for $\gamma$-ray emissions and
LSST/WFST detector for optical afterglows. In addition, we find these events
have the redshifts $z\lesssim 2$ and the inclination angles $\iota\lesssim
20^{\circ}$. These results justify the rough estimation in previous works.
Considering these events as standard sirens to constrain the equation-of-state
parameters of dark energy $w_{0}$ and $w_{a}$, we obtain the potential
constraints of $\Delta w_{0}\simeq 0.02-0.05$ and $\Delta w_{a}\simeq 0.1-0.4$.
