To facilitate the seamless integration of renewable energy, bipolar DC distribution networks (Bi-DCDNs) have been widely adopted in various applications, including the Shenzhen Future Building, the Boeing 787 aircraft, and DC LED lighting systems in Singapore. Bi-DCDNs incorporate diverse flexible devices to improve both economic efficiency and security. However, a comprehensive coordination framework considering the regulatory heterogeneity among these flexible devices remains absent. Hence, this paper proposes a hierarchical coordination framework for flexible devices in Bi-DCDNs. More specifically, the upper-level model considers the operational differences of the DC transformer (DCT) in various modes to determine the optimal switching scheme for reducing losses; In the lower-level model, the control parameters of the DCT, energy storage systems (ESSs), and DC electrical springs (DCESs) are coordinated to enhance voltage quality. Furthermore, to accurately capture the steady-state behavior of flexible devices, the hierarchical framework incorporates the Newton-Raphson power flow method. This method formulates a steady-state model for multiple flexible devices and demonstrates the impact of different control modes of DCT on power flow. Subsequently, a genetic algorithm is used to solve the proposed model, ensuring that suboptimal decisions made at the upper level are rectified at the lower level, and vice versa. The numerical results indicate that the proposed framework achieves the optimal operation for both reduced losses and enhanced voltage quality in Bi-DCDNs. Furthermore, it exhibits advantageous applications for Bi-DCDNs with additional DCTs for remote residential communities.