Distributed energy resources (DERs) in distribution systems offer substantial flexibility potential for power system operations, necessitating effective quantification and optimization methods for their utilization. This paper presents a unified timecoupled flexibility cost quantification approach that aligns with time-coupled flexibility models of various types of DER, whether individual or aggregated. The superior capability in accurately capturing the user inconvenience costs compared to traditional power-range-based, time-decoupled methods is demonstrated, thereby enhancing user willingness to provide flexibility to support the power system operation. Based on the quantification method, a DSO optimization model that activates DER flexibility for energy arbitrage and ancillary services provision in the transmission system while maintaining the distribution system's security and economic efficiency, i.e., TSO-DSO coordination, is developed. The optimization model introduces the concept of marginal flexibility prices (MFPs), which quantify the marginal contribution of flexibility in TSO-DSO coordination. Using MFPs for revenue allocation, incentive compatibility, revenue adequacy, and the non-profit status of DSO are guaranteed, enabling the proposed flexibility provision framework to operate sustainably. Numerical tests on the IEEE test system verify the proposed methods.