Given the uncertainty from renewable production, local loads and battery operating states in microgrid, it is vital to develop an efficient energy management scheme to improve system economics and enhance grid reliability. In this paper, we consider a renewable integrated microgrid scenario including an energy storage system (ESS), bidirectional energy flow from/to conventional power grid and ESS health estimation. We develop a novel demand response-based scheme for microgrid energy management with a long short-term memory (LSTM) network and reinforcement learning (RL), aiming to improve the system operating profit from energy-trading and reduce the battery health cost from energy-scheduling. Specifically, to overcome the uncertainty from future, we utilize LSTM to forecast the unknown demand and electricity price. To obtain the desired ESS control scheme, we apply RL to learn an optimal energy-scheduling strategy. To improve the critical performance of the RL paradigm, we propose a random greedy strategy to encourage exploration. Numerical results show that our proposed algorithm outperforms the baselines by improve the system operating profit by 8.27% and 17.31% while ensuring ESS operating safety. By integrating energy efficiency with sustainable energy management practices, our scheme contributes to long-term environmental and economic resilience.