Engineering self-adaptive systems for software applications necessitates accurate predictions about the state of the underlying application. These predictions can then be used to enable automated cloud operations, such as scaling services in microservices architectures. However, designing an effective self-adaptive system for software applications requires simultaneous predictions across multiple dimensions, including performance, anomalies, and their root causes. While numerous algorithms have been proposed to address performance prediction and anomaly detection, these models typically focus on a single dimension. In this paper, we propose SystemLENS, a novel approach that integrates performance prediction, anomaly detection, and root-cause localization within a unified framework for microservice applications. SystemLENS utilizes Graph Neural Networks (GNNs) and Gated Recurrent Units (GRUs) to first predict latency distributions for traces and the microservice calls involved in generating those traces. These latency distributions are further processed to identify trace-based anomalies and their root causes. By consolidating these tasks into a single model, SystemLENS facilitates comprehensive system monitoring with improved correlations between predictions. We evaluate SystemLENS on benchmark datasets from the domains of performance modeling and anomaly detection, demonstrating its effectiveness in providing an integrated and proactive monitoring solution.