To address high data traffic demands of sixth-generation (6G) networks, this paper proposes a novel architecture that integrates autonomous aerial vehicles (AAVs) and multi-functional reconfigurable intelligent surfaces (MF-RISs) as AM-RIS in fluid antenna (FA)-assisted full-duplex (FD) networks. The AM-RIS provides hybrid functionalities, including signal reflection, amplification, and energy harvesting (EH), potentially improving both signal coverage and sustainability. Meanwhile, FA facilitates fine-grained spatial adaptability at FD-enabled base station (BS), which complements residual self-interference (SI) suppression. We aim at maximizing the overall energy efficiency (EE) by jointly optimizing transmit DL beamforming at BS, UL user power, configuration of AM-RIS, and positions of the FA and AM-RIS. Owing to the hybrid continuous-discrete parameters and high dimensionality of the intractable problem, we have conceived a self-optimized multi-agent hybrid deep reinforcement learning (DRL) framework (SOHRL), which integrates multi-agent deep Q-networks (DQN) and multi-agent proximal policy optimization (PPO), respectively handling discrete and continuous actions. To enhance self-adaptability, an attention-driven state representation and meta-level hyperparameter optimization are incorporated, enabling multi-agents to autonomously adjust learning hyperparameters. Simulation results validate the effectiveness of the proposed AM-RIS-enabled FA-aided FD networks empowered by SOHRL algorithm. The results reveal that SOHRL outperforms benchmarks of the case without attention mechanism and conventional hybrid/multi-agent/standalone DRL. Moreover, AM-RIS in FD achieves the highest EE compared to half-duplex, conventional rigid antenna arrays, partial EH, and conventional RIS without amplification, highlighting its potential as a compelling solution for EE-aware wireless networks.