This dissertation is split into three distinct scientific/mathematical endeavours, each of which involves the study of qualitative dynamics within microfluidic systems in varying degrees and contexts. Chapter 1 involves the unification and extension of dynamical theories connected to the phenomenon of deterministic lateral displacement in microfluidic devices in limits where the size of obstacles in said devices is vanishingly small, and details our use of this extension to develop algorithms that automatically design such devices for prescribed engineering applications in ways that considerably improve over previously reported designs in the literature. Chapter 2 contains work on the design, fabrication and validation of a proof-of-concept microfluidic device to enable high-throughput experimental analyses of non-equilibrium chemical reactions via the rapid and continuous formation of chemically customizable picoliter reactor droplets. Chapter 3 is purely mathematical work centered on establishing the mutual equivalence of all standard chaotic properties in discrete dynamical systems that behave in a sufficiently similar way to sets of infinite symbol sequences under shifting, when the set of infinite symbol sequences is sufficiently "well-posed" in a scientific sense. These results are formulated on sets that are not necessarily shift-invariant but include important subsets of these, and therefore represent an extension of results in that field.