This paper studies cell-free massive multiple-input multiple-output (CF-mMIMO) systems that underpin simultaneous wireless information and power transfer (SWIPT) for separate information users (IUs) and energy users (EUs) in Internet of Things (IoT) networks. We propose a joint access point (AP) operation mode selection and power control design, wherein certain APs are designated for energy transmission to EUs, while others are dedicated to information transmission to IUs. The performance of the system, from both a spectral efficiency (SE) and energy efficiency (EE) perspective, is comprehensively analyzed. Specifically, we formulate two mixed-integer nonconvex optimization problems for maximizing the average sum-SE and EE, under realistic power consumption models and constraints on the minimum individual SE requirements for individual IUs, minimum HE for individual EUs, and maximum transmit power at each AP. The challenging optimization problems are solved using successive convex approximation (SCA) techniques. The proposed framework design is further applied to the average sum-HE maximization and energy harvesting fairness problems. Our numerical results demonstrate that the proposed joint AP operation mode selection and power control algorithm can achieve EE performance gains of up to $4$-fold and $5$-fold over random AP operation mode selection, with and without power control respectively.