  Galaxy cluster mergers are rich sources of information to test cluster
astrophysics and cosmology. However, cluster mergers produce complex projected
signals that are difficult to interpret physically from individual
observational probes. Multi-probe constraints on the gas and dark matter
cluster components are necessary to infer merger parameters that are otherwise
degenerate. We present ICM-SHOX (Improved Constraints on Mergers with SZ,
Hydrodynamical simulations, Optical, and X-ray), a systematic framework to
jointly infer multiple merger parameters quantitatively via a pipeline that
directly compares a novel combination of multi-probe observables to mock
observables derived from hydrodynamical simulations. We report a first
application of the ICM-SHOX pipeline to MACS J0018.5+1626, wherein we
systematically examine simulated snapshots characterized by a wide range of
initial parameters to constrain the MACS J0018.5+1626 merger geometry. We
constrain the epoch of MACS J0018.5+1626 to the range $0$--$60$ Myr
post-pericenter passage, and the viewing angle is inclined $\approx 27$--$40$
degrees from the merger axis. We obtain constraints for the impact parameter
($\lesssim 250$ kpc), mass ratio ($\approx 1.5$--$3.0$), and initial relative
velocity when the clusters are separated by 3 Mpc ($\approx 1700$--3000 km
s$^{-1}$). The primary and secondary clusters initially (at 3 Mpc) have gas
distributions that are moderately and strongly disturbed, respectively. We
discover a velocity space decoupling of the dark matter and gas distributions
in MACS J0018.5+1626, traced by cluster-member galaxy velocities and the
kinematic Sunyaev-Zel'dovich effect, respectively. Our simulations indicate
this decoupling is dependent on the different collisional properties of the two
distributions for particular merger epochs, geometries, and viewing angles.
