We first establish a simple yet powerful necessary and sufficient condition for a binary linear code to be SO, leading to a complete characterization of singly-even codes in this family. We further derive necessary and sufficient conditions on Boolean and vectorial Boolean functions for generating such codes via a standard construction method. Building on this foundation, we propose three general frameworks for constructing binary SO singly-even minimal non-AB linear codes with few weights. The first two approaches are based on designing Boolean and vectorial Boolean functions that simultaneously satisfy multiple conditions. The third method generates new SO codes from existing ones. As a result, we obtain many infinite classes of binary self-orthogonal singly-even minimal linear codes violating the AB condition with few weights and fully determined weight distributions. Particularly, numerical results show that some duals of our codes are optimal or near-optimal.