This paper investigates a simultaneously transmitting and reflecting reconfigurable intelligent surface (STARS)-aided integrated sensing and communication (ISAC) systems in support of full-space energy-efficient data transmissions and target sensing. We formulate an energy efficiency (EE) maximization problem that jointly optimizes a dual-functional radar-communication (DFRC)-empowered base station (BS), considering its ISAC-based active beamforming, along with the passive STARS beamforming configurations of amplitudes, phase shifts, quantization levels, and element selection. Furthermore, relaxed/independent/coupled STARS are considered to examine architectural flexibility. To tackle the non-convex and mixed-integer problem, we propose a joint active-passive beamforming, quantization and element selection (AQUES) scheme based on the alternating optimization: Lagrangian dual and Dinkelbach's transformation tackle fractional equations, whereas successive convex approximation (SCA) convexifies the non-solvable problem; Penalty dual decomposition (PDD) framework and penalty-based convex-concave programming (PCCP) procedure solve amplitude and phase-shifts with the equality constraint; Heuristic search iteratively decides the optimal quantization level; Integer relaxation deals with the discrete element selection variables. Simulation results demonstrate that STARS-ISAC with the proposed AQUES scheme significantly enhances EE while meeting communication rates and sensing quality requirements. The coupled STARS further highlights its superior EE performance over independent and relaxed STARS thanks to its reduced hardware complexity. Moreover, AQUES outperforms existing configurations and benchmark methods in the open literature across various network parameters and deployment scenarios.