The use of Natural Language Processing techniques in the legal domain has become established for supporting attorneys and domain experts in content retrieval and decision-making. However, understanding the legal text poses relevant challenges in the recognition of domain-specific entities and the adaptation and explanation of predictive models. This paper addresses the Legal Entity Name Recognition (L-NER) and Court judgment Prediction (CPJ) and Explanation (CJPE) tasks. The L-NER solution explores the use of various transformer-based models, including an entity-aware method attending domain-specific entities. The CJPE proposed method relies on hierarchical BERT-based classifiers combined with local input attribution explainers. We propose a broad comparison of eXplainable AI methodologies along with a novel approach based on NER. For the L-NER task, the experimental results remark on the importance of domain-specific pre-training. For CJP our lightweight solution shows performance in line with existing approaches, and our NER-boosted explanations show promising CJPE results in terms of the conciseness of the prediction explanations.