Prognostic markers such as overall survival (OS) and tertiary lymphoid structure (TLS) ratios, alongside diagnostic signatures like primary cancer-type classification, provide critical information for treatment selection, risk stratification, and longitudinal care planning across the oncology continuum. However, extracting these signals solely from sparse, high-dimensional multi-omics data remains a major challenge due to heterogeneity and frequent missingness in patient profiles. To address this challenge, we present SeNMo, a self-normalizing deep neural network trained on five heterogeneous omics layers-gene expression, DNA methylation, miRNA abundance, somatic mutations, and protein expression-along with the clinical variables, that learns a unified representation robust to missing modalities. Trained on more than 10,000 patient profiles across 32 tumor types from The Cancer Genome Atlas (TCGA), SeNMo provides a baseline that can be readily fine-tuned for diverse downstream tasks. On a held-out TCGA test set, the model achieved a concordance index of 0.758 for OS prediction, while external evaluation yielded 0.73 on the CPTAC lung squamous cell carcinoma cohort and 0.66 on an independent 108-patient Moffitt Cancer Center cohort. Furthermore, on Moffitt's cohort, baseline SeNMo fine-tuned for TLS ratio prediction aligned with expert annotations ( < 0.05) and sharply separated high- versus low-TLS groups, reflecting distinct survival outcomes. Without altering the backbone, a single linear head classified primary cancer type with 99.8% accuracy across the 33 classes. By unifying diagnostic and prognostic predictions in a modality-robust architecture, SeNMo demonstrated strong performance across multiple clinically relevant tasks, including survival estimation, cancer classification, and TLS ratio prediction, highlighting its translational potential for multi-omics oncology applications.