This paper presents a novel multi-channel cooperative spectrum sensing and scheduling (MC$_2$S$_3$) framework for spectrum and energy harvesting cognitive Internet of Things (IoT) networks. We address the challenge of maximizing network throughput by formulating a combinatorial problem that jointly optimizes the sensing scheduling of primary channels (PCs), the assignment of IoT devices for sensing scheduled PCs, and the clustering and allocation of IoT nodes to efficiently use discovered idle PCs; subject to spectrum utilization and collision avoidance constraints. Recognizing the inherent complexity of the underlying NP-hard mixed-integer non-linear programming (MINLP) problem, we propose a decomposition strategy that decouples the master problem into PC exploration and exploitation sub-problems. In the exploration phase, we derive closed-form solutions for optimal sensing durations and detection thresholds that satisfies spectrum utilization and collision avoidance constraints, which are then used to develop a priority metric to rank PCs. The proposed PC ranking informs a sequential PC scheduling and IoT sensing assignment approach that exploits a linear bottleneck assignment (LBA) strategy, proceeding until further scheduling does not enhance network utility. For the exploitation phase, we leverage a non-orthogonal multiple access (NOMA) strategy to multiplex multiple IoT nodes on a single PC, employing an iterative linear sum assignment (LSA) method for efficient allocation to maximize utilization of idle PCs. Numerical results validate the efficacy of our proposed methodologies, reaching an accuracy of approximately 99% in the order of milliseconds, significantly outperforming time complexity of brute-force benchmarks.