We outline a comprehensive computational physics-based investigation of droplet generation characteristics within a double inlet microfluidic T-junction with a semicylindrical obstacle. The interaction of continuous and dispersed fluids triggered by obstacle radius, obstacle position, and the capillary number on the droplet generation is explored in detail. Finite element-based level-set formalism is adopted to track the interface of the two phases in a transient 3D framework. Emphasis has been put to identify the suitable geometrical orientation of the microfluidic confinement for yielding fine spherical droplets with a faster generation rate. The interactions between the pressure forces developed across the obstacle and the amount of continuous fluid striking the dispersed fluid govern the pinch-off phenomenon to yield droplets. The study reveals that the confinement with a larger obstacle radius is susceptible to form fine spherical droplets with a faster generation rate and the production is significantly influenced by the obstacle position. For higher capillary numbers, the dispersed phase goes through extensive elongation before the rupture.