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Active colloidal systems with nonequilibrium self-organization constitute a long- standing, challenging area in material sciences and biology. To understand how hydrodynamic flow may be used to actively control self-assembly of Janus particles (JPs), we developed a model for the many-body hydrodynamics of amphiphilic JPs suspended in a viscous fluid with imposed far-field background flows [Fu et al., J. Fluid Mech. 941, A41 (2022)]. In this paper we alter the hydrophobic distribution on the JP-solvent interface to investigate the hydrodynamics that underlies the various morphologies and rheological properties of the JP assembly in the suspension. We find that JPs assemble into unilamellar, multilamellar, and striated structures. To introduce dynamics, we include a planar linear shear flow and a steady Taylor-Green mixing flow and measure the collective dynamics of JP particles in terms of their (a) free energy from the hydrophobic interactions between the JPs, (b) order parameter for the ordering of JPs in terms of alignment of their directors, and (c) strain parameter that captures the deformation in the assembly. We characterize the effective material properties of the JP structures and find that the unilamellar structure increases orientation order under shear flow, the multilamellar structure behaves as a shear thinning fluid, and the striated structure possesses a yield stress. These numerical results provide insights into dynamic control of nonequilibrium active biological systems with similar self-organization.