Using computer simulations of a soft, coarse-grained model and a Lennard-Jones bead-spring model, we investigate the behavior of random block copolymer (RBCP) blends in the bulk and in the vicinity of a solid substrate. The RBCP is comprised of six random sequences of A and B blocks and adopts a microemulsion-like morphology as we increase the incompatibility between A and B segment species. The soft, coarse-grained model is used to efficiently equilibrate the morphology and molecular conformations in the melt, and its explicit molecular conformations are used to generate equilibrated starting configurations for the bead spring model. Since the pure A and B melts are structurally asymmetric, A-rich and B-rich domains differ in their density and viscosity in the melt or mechanical properties in the glassy state, respectively. We demonstrate that the self-assembled morphology in the melt gives rise to a viscoelastic transient plateau in the stress autocorrelation function. While an analysis of the entanglement density reveals a slight increase of the number of entanglements in response to microphase separation, the viscoelasticity of the RBCP blend chiefly stems from the slow morphological relaxation and the transient trapping of blocks inside domains. Upon quenching the microphase-separated structure below the glass transition temperature, the shear modulus increases about an order of magnitude compared to the viscoelastic plateau. The structural asymmetry of the segment species gives rise to spatially heterogeneous, local bulk and shear moduli that correlate with local composition fluctuations. The vicinity of a solid substrate that prefers one segment species gives rise to variations of the composition that propagate about three molecular extensions into the bulk. In this extended interphase, we observe an oscillatory decay of the composition and the local mechanical properties.