If Chapter 1 is a systematic literature review (QUANTS/QUALS) while Chapters 2, 3, and 4 are empirical chapters (QUALS), is this considered a mixed methods research design?
No, it is not mixed methods. Literature review is background of any research but not a research in itself until you drive some finding and it's use and implications. The technique use to develop some finding is considered method.
No, you are purely doing a qualitative study. Mixed-methods studies demand that you take empirical data (primary) using both quant and qual approaches.