Cooperative perception (CP) is attracting increasing attention and is regarded as the core foundation to support cooperative driving automation, a potential key solution to addressing the safety, mobility, and sustainability issues of contemporary transportation systems. However, current research on CP is still at the beginning stages where a systematic problem formulation of CP is still missing, acting as the essential guideline of the system design of a CP system under real-world situations. In this paper, we formulate a universal CP system into an optimization problem and a mobile-edge-cloud framework called Cooperverse. This system addresses CP in a mixed connectivity and automation environment. A Dynamic Feature Sharing (DFS) methodology is introduced to support this CP system under certain constraints and a Random Priority Filtering (RPF) method is proposed to conduct DFS with high performance. Experiments have been conducted based on a high-fidelity CP platform, and the results show that the Cooperverse framework is effective for dynamic node engagement and the proposed DFS methodology can improve system CP performance by 14.5% and the RPF method can reduce the communication cost for mobile nodes by 90% with only 1.7% drop for average precision.