Integrated energy systems (IESs) are complex systems consisting of diverse operating units spanning multiple domains. To address its operational challenges, we propose a physics-informed hybrid time-series neural network (NN) surrogate to predict the dynamic performance of IESs across multiple time scales. This neural network-based modeling approach develops time-series multi-layer perceptrons (MLPs) for the operating units and integrates them with prior process knowledge about system structure and fundamental dynamics. This integration forms three hybrid NNs (long-term, slow, and fast MLPs) that predict the entire system dynamics across multiple time scales. Leveraging these MLPs, we design an NN-based scheduler and an NN-based economic model predictive control (NEMPC) framework to meet global operational requirements: rapid electrical power responsiveness to operators requests, adequate cooling supply to customers, and increased system profitability, while addressing the dynamic time-scale multiplicity present in IESs. The proposed day-ahead scheduler is formulated using the ReLU network-based MLP, which effectively represents IES performance under a broad range of conditions from a long-term perspective. The scheduler is then exactly recast into a mixed-integer linear programming problem for efficient evaluation. The real-time NEMPC, based on slow and fast MLPs, comprises two sequential distributed control agents: a slow NEMPC for the cooling-dominant subsystem with slower transient responses and a fast NEMPC for the power-dominant subsystem with faster responses. Extensive simulations demonstrate that the developed scheduler and NEMPC schemes outperform their respective benchmark scheduler and controller by about 25% and 40%. Together, they enhance overall system performance by over 70% compared to benchmark approaches.