The exponential growth of mobile data usage and video streaming has required advancements in network technology to meet the increasing demand for high-quality, real-time media delivery. Cloud computing has been proposed as a primary solution for processing and storing vast amounts of network-generated data. However, this approach requires significant data transmission, degrading network performance and challenging real-time application requirements. Multi-access Edge Computing has emerged to address these issues by bringing computing and storage resources closer to the data source and the Radio Access Network (RAN). MEC enables real-time data processing with lower delay and energy consumption than traditional cloud-based architectures. Lately, the landscape of HTTP Adaptive Streaming (HAS) transformed significantly. Video content consumption surged in popularity, including Video on Demand, live streaming, video sharing, video conferencing, and applications incorporating video technologies like Augmented, Virtual and Extended Reality. By 2027, mobile video traffic is projected to constitute 79 percent of total mobile traffic, highlighting the need for efficient resource allocation in telecommunication networks. MEC offers a solution by storing video content at the network edge, reducing delivery delay and backhaul congestion while enhancing Quality of Service (QoS) and Quality of Experience (QoE). This research introduces novel solutions to optimize resource management in MEC-based networks. The The Multi-access Edge Computing-Optimization Problem-Server Allocation (MEC-OP-SA) algorithm proposes a cross-layer joint optimization model leveraging MEC and Server and Network Assisted Dynamic Adaptive Streaming over HTTP (SAND-DASH) to balance QoE, fairness, and system utilization while addressing radio resource constraints. The Cross-Layer QoE-Driven Bitrate Allocation (CLQDBA) algorithm introduces a low-complexity, greedy-based method to improve system utilization, maximize QoE, and reduce backhaul traffic by caching popular videos. The MEC Collaborative Cross-Layer Bitrate Allocation (MCCBA) algorithm enhances QoE through collaboration between MEC servers and RAN components, addressing resource allocation challenges while improving fairness. Additionally, this study provides a comprehensive survey of current solutions, trends, and open issues in MEC and HAS research. These contributions are rigorously evaluated through simulations, confirming significant improvements in QoE, fairness, and system performance across diverse scenarios.