We investigate a multi-low Earth orbit (LEO) satellite system that simultaneously provides positioning and communication services to terrestrial user terminals. To address the challenges of accurately acquiring channel state information in LEO satellite systems, we propose a novel two-timescale positioning-aided channel estimation framework, exploiting the distinct variation rates of position-related parameters and channel gains inherent in LEO satellite channels. Using the misspecified Cramér-Rao bound (MCRB) theory, we systematically analyze positioning performance under practical imperfections, such as inter-satellite clock bias and carrier frequency offset. Furthermore, we theoretically demonstrate how position information derived from downlink positioning can enhance uplink channel estimation accuracy, even in the presence of positioning errors, through an MCRB-based analysis. To address the limited link budgets and communication rates of single-satellite communication, we develop a multi-LEO cooperative beamforming strategy for downlink transmission that leverages cluster-wise satellite cooperation while maintaining reduced complexity. Theoretical analyses and numerical results confirm the effectiveness of the proposed framework in facilitating high-precision downlink positioning under practical imperfections, facilitating uplink channel estimation, and enabling efficient downlink communication.