The theory of rough sets produces a potent framework for administrating uncertainty and ambiguity in data, which is crucial for effective decision-making. However, the reliance on equivalence relations within this framework has led to the exploration of various generalizations and extensions. In this paper, we introduce eight new types of initial neighborhoods, expanding on the idea of initial neighborhoods, and examine the relationships and properties of twelve distinct types of neighborhoods derived from binary relations. We define initial-minimal and initial-maximal neighborhoods and develop eight types of rough approximations (-approximations) that generalize Pawlak's theory. These new approximations significantly improve upon previous methods, achieving accuracy rates of up to 100%. Furthermore, we implement Generalized Nano-topological frameworks in conjunction with our novel methodologies to address clinical applications, particularly focusing on advancing diagnostic strategies for Covid-19. By employing a universal binary relation, we clarify the effectiveness for our methodology per enhancing decision-making processes and pinpointing significant risk factors associated with Covid-19. Additionally, we introduce two algorithms for decision-making problems in information systems, emphasizing the broader applicability and significance of our approach across various fields.