Cooperative perception is critical for autonomous driving, overcoming the inherent limitations of a single vehicle, such as occlusions and constrained fields-of-view. However, current approaches sharing dense Bird's-Eye-View (BEV) features are constrained by quadratically-scaling communication costs and the lack of flexibility and interpretability for precise alignment across asynchronous or disparate viewpoints. While emerging sparse query-based methods offer an alternative, they often suffer from inadequate geometric representations, suboptimal fusion strategies, and training instability. In this paper, we propose SparseCoop, a fully sparse cooperative perception framework for 3D detection and tracking that completely discards intermediate BEV representations. Our framework is built on a trio of innovations designed for robust and efficient fusion: a kinematic-grounded instance query that uses an explicit state vector with 3D geometry and velocity for precise spatio-temporal alignment; a coarse-to-fine aggregation module that effectively integrates information from both matched and unmatched instances; and a cooperative instance denoising task that provides stable, abundant supervision to accelerate and stabilize training. Experiments on the V2X-Seq and Griffin datasets demonstrate that SparseCoop achieves new state-of-the-art performance in both 3D detection and tracking. Notably, it delivers this performance with superior computational efficiency and a highly competitive transmission cost, while showing remarkable robustness to real-world challenges like communication latency.