Vertical-external-cavity surface-emitting lasers (VECSELs) have been successfully used in the visible and near-infrared to achieve high output power with excellent Gaussian beam quality. However, the concept of VECSEL has been impossible to implement for quantum-cascade (QC) lasers due to the "intersubband selection rule". We have recently demonstrated the first VECSEL in the terahertz range. The enabling component for the QC-VECSEL is an amplifying metasurface reflector composed of a sparse array of metallic sub-cavities, which allows the normally incident radiation to interact with the electrically pumped QC gain medium. In this work, we presented multiple design variations based on the first demonstrated THz QC-VECSEL, regarding the lasing frequencies, the output coupler and the intra-cavity aperture. Our work on THz QC-VECSEL initiates a new approach towards achieving scalable output power in combination with a diffraction-limited beam pattern for THz QC-lasers. The design variations presented in this work further demonstrate the practicality and potential of VECSEL approach to make ideal terahertz QC-laser sources.