In this paper, we study the intricate realm of digital twin synchronization and deployment in multi-access edge computing (MEC) networks, with the aim of optimizing and balancing the two performance metrics Age of Information (AoI) and energy efficiency. We jointly consider the problems of edge association, power allocation, and digital twin deployment. However, the inherent randomness of the problem presents a significant challenge in identifying an optimal solution. To address this, we first analyze the feasibility conditions of the optimization problem. We then examine a specific scenario involving a static channel and propose a cyclic scheduling scheme. This enables us to derive the sum AoI in closed form. As a result, the joint optimization problem of edge association and power control is solved optimally by finding a minimum weight perfect matching. Moreover, we examine the one-shot optimization problem in the contexts of both frequent digital twin migrations and fixed digital twin deployments, and propose an efficient online algorithm to address the general optimization problem. This algorithm effectively reduces system costs by balancing frequent migrations and fixed deployments. Numerical results demonstrate the effectiveness of our proposed scheme in terms of low cost and high efficiency.