This study investigates the stability and performance of mixed-traffic flows consisting of human-driven vehicles (HDVs), connected autonomous vehicles (CAVs), autonomous vehicles (AVs), and connected human-driven vehicles (CHVs). Recognizing the complexity introduced by multi-vehicle interactions in such heterogeneous traffic, a refined CAV car-following model that integrates multi-vehicle state information, including headway, weighted velocity differences, weighted acceleration, and optimal velocity memory effects from both front and rear vehicles, is introduced. Through theoretical analysis of the model's linear and nonlinear stability, the key parameters that enhance flow stability in mixed environments are determined. Numerical simulations across braking, start-up, and ring road scenarios validate the proposed model's efficacy, demonstrating that it can effectively suppress traffic congestion and reduce oscillations, thereby improving traffic flow stability. This work offers valuable insights into the behavior of connected vehicles within mixed traffic and highlights the potential for CAV-based strategies to enhance both safety and efficiency in future transportation systems.