To address the challenges in the collaborative planning of data centers into the low-carbon distribution networks under the computing-electricity joint market, this paper proposes a multi-objective interval-stochastic optimization approach. By introducing a computing power leasing mechanism, the method optimizes resource utilization efficiency and economic benefits while reducing operational costs and carbon emissions. To tackle the multi-objective optimization and high-dimensional uncertainty issues in the collaborative planning process, an adaptive constraint- handling interval multi-objective evolutionary algorithm based on decomposition with two crossover strategies (ACIMOEA/D-TCS) is designed. This algorithm efficiently solves the Pareto frontier and provides robust and feasible optimization solutions. The results show that the participation of data centers in the computing power market significantly improves resource utilization efficiency and economic benefits, while also effectively reducing carbon emissions. Through the optimization of computing power resource leasing and distribution system operation, the proposed model achieves significant improvements in both economic and environmental benefits, providing new theoretical methods and solutions for collaborative planning in the computing-electricity joint market.