Microwave linear analog computers (MiLACs) offer a transformative paradigm for future multiple-input multiple-output (MIMO) systems by shifting complex signal processing into the analog domain, thereby significantly reducing computational complexity, radio-frequency chains, and analog-digital converters, while speeding up computation. However, the practical deployment of MiLACs is severely constrained by the inherent hardware losses of the tunable admittance components (TACs) interconnecting MiLAC ports, which introduce severe inter-stream interference and fundamentally limit the spectral efficiency (SE) of the system. In addition, while denser architectures offer greater spatial degrees of freedom to mitigate inter-stream interference, the cumulative hardware losses and power consumption of massive TACs severely degrade the system's energy efficiency (EE). Consequently, designing architectures for lossy MiLACs emerges as a critical yet unresolved challenge, as it necessitates striking a delicate tradeoff between interference suppression and cumulative hardware losses/power consumption. To address this challenge, this paper investigates the joint MiLAC architecture design and performance (SE/EE) maximization in lossy MiLAC-aided MIMO systems. We propose a novel learning-based joint architecture and performance optimization framework (LJAPOF) that unifies the design of MiLAC architectures and analog beamforming configurations for lossy MiLACs under both SE- and EE-oriented objectives. Numerical results demonstrate that by intelligently navigating the fundamental tradeoff between interference suppression and hardware/power consumption, the proposed LJAPOF can design optimal MiLAC architectures that consistently outperform stem-connected and fully-connected MiLACs in maximizing the system's SE and EE.