Viscoelastic materials such as rubbers and polymers are widely employed in engineering applications requiring energy dissipation and vibration damping. This work presents a comprehensive topology optimization framework for multi-material viscoelastic structures, addressing two critical design scenarios: impact mitigation under transient loading and thermo-mechanical coupled design under thermal environments. The proposed methodology employs the Generalized Maxwell model to capture time-dependent stress relaxation and energy dissipation mechanisms inherent to viscoelastic materials. The transient response is solved using the Newmark-beta method in the time domain, enabling explicit consideration of path-dependent stress evolution, arbitrary loading histories, and initial conditions. A multi-phase material interpolation scheme based on the Rational Approximation of Material Properties (RAMP) is implemented to represent distinct viscoelastic phases efficiently. For impact mitigation applications, we formulate optimization problems to minimize reaction forces and peak accelerations using a p-norm approach. Numerical examples demonstrate that optimized multi-material designs significantly outperform single-material configurations by creating effective buffering mechanisms through the strategic spatial distribution of stiff and soft phases. For thermo-viscoelastic applications, we incorporate Arrhenius-based temperature-dependent relaxation times within a weakly-coupled thermal-mechanical formulation. The framework addresses both displacement minimization and energy dissipation maximization objectives. Results reveal objective dependent design strategies: displacement minimization concentrates stiff material in hot regions to compensate thermal softening, while dissipation maximization places stiff material in cold regions to preserve stress generation capacity. Adjoint sensitivity analysis is derived using the discretize-then-differentiate approach, accounting for recursive Maxwell stress dependencies and temperature field effects. The methodology is verified through finite difference comparisons and validated across multiple numerical benchmarks. This work establishes a unified framework for time- and temperature-dependent viscoelastic topology optimization with broad applicability to automotive components, seismic isolation devices, and precision machinery.