Materials properties has been one limiting factor for electronic device performances. The “origin” of materials properties or how they respond to outside stimulus, comes from its structure, which is eventually determined by processing conditions. Among those endeavors of engineering emergent Physics phenomena, utilizing magnetic topological insulators (MTI) and quantum anomalous Hall effect (QAHE) is one prospectus direction for beyond-CMOS (complementary metal-oxide-semiconductor) technologies. Molecular beam epitaxy (MBE) is one state-of-the-art method for growing single crystalline thin film MTI. Yet the underlying materials science in controlling quantization quality is still not fully developed in this newly raised field. In this dissertation, I am mainly addressing some structure-property-processing relationships that are not raised up previously during MBE growth of MTI (mainly Cr:(BixSb1−x)2Te3 (CBST) here) and its heterostructures, with characterization methods also playing an essential role. Similar to many other van der Waals chalcogenides, the growth of CBST by MBE is widely tuned into “adsorption control” mode. But there is no explicit experiments showing this optimized growth condition. In this dissertation, it is the first-time clear mapping of MTI growth temperatures and an optimized growth mode, using both growth rate and anomalous Hall voltages as evaluations. With quasi van der Waals Epitaxy (qvdWE), the growth of CBST on GaAs (111) or some III-V substrates is distinguished from other substrates, which is not well noticed despite the fact that its advantages have been used. Here in this dissertation, I have for the first time shown the features of both coherent interfaces and the existence of strain originating from qvdWE at the same time. The application of topological insulators (TI) as epitaxial buffer layers for other quantum materials is also examined in this dissertation. By controlling the lattice constant of the TI buffer layers, we have accidentally tuned the phases of the MnTe grown on top. This can be utilized for better enhancing the quality of epitaxial MnTe for its potential usage in future spintronic devices. These understandings in the processing conditions are important for not only future MTI device performances, but is also pushing the boundary of fundamental materials science.