For scene understanding in unstructured environments, an accurate and uncertainty-aware metric-semantic mapping is required to enable informed action selection by autonomous systems. Existing mapping methods often suffer from overconfident semantic predictions, and sparse and noisy depth sensing, leading to inconsistent map representations. In this paper, we therefore introduce EvidMTL, a multi-task learning framework that uses evidential heads for depth estimation and semantic segmentation, enabling uncertainty-aware inference from monocular RGB images. To enable uncertainty-calibrated evidential multi-task learning, we propose a novel evidential depth loss function that jointly optimizes the belief strength of the depth prediction in conjunction with evidential segmentation loss. Building on this, we present EvidKimera, an uncertainty-aware semantic surface mapping framework, which uses evidential depth and semantics prediction for improved 3D metric-semantic consistency. We train and evaluate EvidMTL on the NYUDepthV2 and assess its zero-shot performance on ScanNetV2, demonstrating superior uncertainty estimation compared to conventional approaches while maintaining comparable depth estimation and semantic segmentation. In zero-shot mapping tests on ScanNetV2, EvidKimera outperforms Kimera in semantic surface mapping accuracy and consistency, highlighting the benefits of uncertainty-aware mapping and underscoring its potential for real-world robotic applications.