The stringent link budget, caused by long propagation distances and payload constraints, poses a fundamental bottleneck for single-satellite transmission. Although LEO mega-constellations make multi-satellite cooperative transmission (MSCT), such as distributed precoding (DP), increasingly feasible, its cooperative gains critically rely on stringent time-frequency-phase synchronization (TFP-Sync), which is difficult to maintain under rapid channel variation and feedback latency. To address this issue, this paper proposes a joint CSI acquisition, feedback, and phase-level synchronization (JCAFPS) framework for MSCT. Specifically, to enable reliable, overhead-efficient CSI acquisition, we design a beam-domain adjustable phase-shift tracking reference signal (TRS) transmission scheme, along with criteria for the TRS and CSI-feedback periods. Then, exploiting deterministic orbital motion and dominant LoS propagation, we establish a polynomial model for the temporal evolution of delay and Doppler shift, and derive an OFDM-based multi-satellite signal model under non-ideal synchronization. The analysis reveals that, unlike the single-satellite case, the composite multi-satellite channel exhibits nonlinear time-frequency-varying phase behavior, necessitating symbol- and subcarrier-wise phase precompensation for coherent transmission. Based on these results, we develop a practical closed-loop realization integrating single-TRS-based channel parameter estimation, multi-TRS-based channel prediction, predictive CSI feedback, and user-specific TFP precompensation. Numerical results demonstrate that the proposed framework achieves accurate CSI acquisition and precise TFP-Sync, enabling DP-based dual-satellite cooperative transmission to approach the theoretical 6 dB power gain over single-satellite transmission, while remaining robust under extended prediction durations and enlarged TRS periods.