Quantum heat transport governs the energy exchange processes and statistical laws in non-equilibrium quantum systems, and also plays a pivotal role in the advance of quantum thermodynamics. In this work, we comprehensively investigate steady-state thermal transport properties of a noncommuting coupled spin system driven by a finite temperature bias. The system comprises interacting spin ensembles separately coupled to independent bosonic thermal reservoirs. We employ the quantum dressed master equation approach within the framework of open quantum system theory to properly analyze the non-equilibrium dynamics, ensuring the validity of the transport results in the strong coupling regime. Our results demonstrate that the noncommuting spin coupling serves as a significant resource for modulating nonlinearities of the heat current. Specifically, in the weak spin-coupling regime, the system exhibits a robust negative differential thermal conductance (NDTC) phenomenon across various spin numbers. By deriving analytical expressions of the heat current in both single-spin and large-spin limits, we reveal that this NDTC behavior is dominated by microscopic cycle fluxes. Physically, this arises because the spin excitation channels induced by the cold reservoir are suppressed under the large temperature bias, thereby blocking the energy exchange cycles. Conversely, in the strong spin-coupling and large temperature bias regime, the quantum system demonstrates a significant thermal rectification effect. This high rectification effciency originates from the unidirectional saturation of the heat current, making the system a promising candidate for high-performance thermal diodes. Furthermore, we extend the model to a three-terminal configuration to construct a quantum thermal transistor. By manipulating the temperature of the gate reservoir, we achieve effcient modulation and amplification of the heat flow between the source and the drain. It is shown that the heat amplification factor β_R can far exceed unity in specific operating regions, confirming the realization of significant thermal amplification. These findings not only elucidate the rich nonlinear transport phenomena induced by noncommuting interactions, but also provide a theoretical basis for designing controllable quantum thermal logic devices, such as thermal rectifiers and transistors.