Flow cell arrays (FCAs) concurrently provide efficient on-chip liquid cooling and electrochemical power generation. This technology is especially promising for three-dimensional multi-processor systems-on-chip (3D MPSoCs) realized in deeply scaled technologies, which present very challenging power and thermal requirements. Indeed, FCAs effectively improve power delivery network (PDN) performance, particularly if switched capacitor (SC) converters are employed to decouple the flow cells and the systems-on-chip voltages, allowing each to operate at their optimal point. Nonetheless, the design of FCA-based solutions entails non-obvious considerations and trade-offs, stemming from their dual role in governing both the thermal and power delivery characteristics of 3D MPSoCs. Showcasing them in this paper, we explore multiple FCA design configurations and demonstrate that this technology can decrease the temperature of a heterogeneous 3D MPSoC by 78°C, and its total power consumption by 46%, compared to a high-performance cold-plate based liquid cooling solution. At the same time, FCAs enable up to 90% voltage drop recovery across dies, using SC converters occupying a small fraction of the chip area. Such outcomes provide an opportunity to boost 3D MPSoC computing performance by increasing the operating frequency of dies. Leveraging these results, we introduce a novel temperature and voltage-aware model predictive control (MPC) strategy that optimizes power efficiency during run-time. We achieve application-wide speed-ups of up to 16% on various machine learning (ML), data mining, and other high-performance benchmarks while keeping the 3D MPSoC temperature below 83°C and voltage drops below 5%.