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The rapid production of mobile devices along with the wireless applications boom is continuing to evolve daily. This motivates the exploitation of wireless spectrum using multiple Radio Access Technologies (multi-RAT) and developing innovative network selection techniques to cope with such intensive demand while improving Quality of Service (QoS). Thus, we propose a distributed framework for dynamic network selection at the edge level, and resource allocation at the Radio Access Network (RAN) level, while taking into consideration diverse applications' characteristics. In particular, our framework employs a deep Multi-Agent Reinforcement Learning (DMARL) algorithm, that aims to maximize the edge nodes' quality of experience while extending the battery lifetime of the nodes and leveraging adaptive compression schemes. Indeed, our framework enables data transfer from the network's edge nodes, with multi-RAT capabilities, to the cloud in a cost and energy-efficient manner, while maintaining QoS requirements of different supported applications. Our results depict that our solution outperforms state-of-the-art techniques of network selection in terms of energy consumption, latency, and cost.