The tunability of multifunctional properties of Triply Periodic Minimal Surfaces (TPMSs) is mainly constrained to variations in their shell thickness. To address this limitation, a novel approach inspired by 2D cellular solids is introduced, utilizing discrete conformal mapping of perforated 2D cellular layouts onto TPMS surfaces to minimize distortion during 3D mapping. This study focuses on primitive (P)-type Conformally Perforated Shellular Metamaterials (CPSMs), evaluating their thermomechanical and acoustic properties through computational analyses and experimentation on 3D-printed samples. Effective thermomechanical properties are determined via asymptotic homogenization, which demonstrate that the thermal conductivity of CPSMs can be calculated by multiplying the effective thermal conductivities of the 3D shellular and 2D thermally-isotropic cellular architectures. Thermal conductivity and elastic stiffness are systematically enhanced by engineering the shell's in-plane architecture while preserving the 3D topology. The optimized design demonstrates a significant increase in elastic stiffness compared to an intact P-shellular of the same density. Numerical and experimental results reveal that the 2D-mapped architecture effectively tunes the acoustic bandwidth and bandgap frequency range of CPSMs, achieving a 79% increase in bandwidth and a 34% reduction in material density compared to an intact P-shellular. These findings highlight CPSMs' potential for structural, thermal, and acoustic applications, advancing the development of additively manufactured TPMS-based multifunctional metamaterials.